MONITORING NG PROGRESS ANSWERS Chapter 5 Pacing Guide. 228 Chapter 5. Chapter Opener/ 0.5 Day Mathematical Practices. Chapter Review/ Chapter Tests

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1 MONIOING NG OG NW hapter 5 acing Guide hapter Opener/ 0.5 ay Mathematical ractices ection 1 ection ection 3 ection 4 uiz ection 5 ection 6 ection 7 ection 8 hapter eview/ hapter ests otal hapter 5 Year-to-ate 1.5 ays 1 ay 1 ay ays 0.5 ay 1.5 ays ays 1 ay ays ays 15 ays 63 ays 5 ongruent riangles 5.1 ngles of riangles 5. ongruent olygons 5.3 roving riangle ongruence by 5.4 quilateral and Isosceles riangles 5.5 roving riangle ongruence by 5.6 roving riangle ongruence by and 5.7 Using ongruent riangles 5.8 oordinate roofs the ig Idea Hang Glider (p. 78) Lifeguard ower (p. 55) arn (p. 48) Home ecor (p. 41) ainting (p. 35) 8 hapter 5

2 Laurie s Notes hapter ummary In this chapter, students will work with a variety of proof formats as they investigate triangle congruence. Methods for establishing triangle congruence (,,, and ) are established using rigid motions. he proof of each congruence criteria for triangles is done by composing transformations. his means a sequence of rigid motions maps one triangle onto another triangle. Other proof styles presented in this chapter include the two-column proof, the paragraph or narrative proof, and finally, in the last lesson, the coordinate proof. In addition to working with proofs, there are properties of equilateral and isosceles triangles that are proven. he use of dynamic geometry software for the explorations is highly encouraged. his tool provides students the opportunity to explore and make conjectures, mathematical practices we want to develop in all students. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations eal-life M Videos caffolding in the lassroom Graphic Organizers: Y-hart Y-hart can be used to compare two topics. tudents list differences between the two topics in the branches of the Y and similarities in the base of the Y. Y-hart serves as a good tool for assessing students knowledge of a pair of topics that have subtle but important differences. You can include blank Y-harts on tests or quizzes for this purpose. OMMON O OGION Middle chool Understand that figures are congruent if they can be related by a sequence of translations, reflections, and rotations. raw triangles with given conditions. ind the distance between points with the same x- or y-coordinate. olve two-step equations. lgebra 1 Write equations in one variable. olve linear equations that have variables on both sides. Graph points and lines in the coordinate plane. Use properties of radicals to simplify expressions. Geometry Identify and use corresponding parts. Use theorems about the angles of a triangle. Use,, HL,, and to prove two triangles congruent. rove constructions. Write coordinate proofs. ection tandards ummary ommon ore tate tandards 5.1 Learning HG-O..10, HG-MG Learning HG-O Learning HG-O..8, HG-MG Learning 5.5 Learning 5.6 Learning HG-O Learning HG-..5 HG-O..10, HG-O..13, HG-MG..1 HG-O..8, HG-MG..1, HG-MG Learning HG-G..4 hapter 5-8

3 uestioning in the lassroom What I really want to know is... Have each student write a pertinent question on a 3 5 card. ollect the cards (check the cards before using them), and randomly select a question to ask the class. fter class discussion, the last person to participate randomly selects another question to ask the class. Laurie s Notes Maintaining Mathematical roficiency Using the Midpoint and istance ormulas Have students review the Midpoint ormula, page, and the istance ormula, page 3. OMMON O tudents may subtract instead of adding when using the Midpoint ormula. oint out that the coordinates of the midpoint of a segment are the means (averages) of the coordinates of the endpoints. olving quations with Variable on oth ides emind students that the goal is to get the variable alone on one side of the equation. OMMON O ecause students are undoing an equation, addition and subtraction should be undone before multiplication and divisionthe opposite of the Order of Operations. Mathematical ractices (continued on page 30) he eight Mathematical ractices focus attention on how mathematics is learnedprocess versus content. age 30 demonstrates the need for students to understand that definitions are biconditional statements. lthough a definition may be stated with many fewer words, it can also be written as a biconditional statement. Use the Mathematical ractices page to help students develop mathematical habits of mindhow mathematics can be explored and how mathematics is thought about. If students need help... tudent Journal Maintaining Mathematical roficiency Lesson utorials If students got it... Game loset at igideasmath.com tart the next ection kills eview Handbook -9 hapter 5

4 Maintaining Mathematical roficiency Using the Midpoint and istance ormulas xample 1 he endpoints of are (, 3) and (4, 7). ind the coordinates of the midpoint M. Use the Midpoint ormula. M ( , ) = M (, 10 ) = M(1, 5) he coordinates of the midpoint M are (1, 5). xample ind the distance between (0, 5) and (3, ). = (x x 1 ) + (y y 1 ) istance ormula = (3 0) + [ (5)] ubstitute. = ubtract. = valuate powers. = 58 dd. 7.6 Use a calculator. he distance between (0, 5) and (3, ) is about 7.6. ind the coordinates of the midpoint M of the segment with the given endpoints. hen find the distance between the two points. ommon ore tate tandards 8.G..8 find the distance between two points in a coordinate system. H-I..3 olve linear equations in one variable,. NW 1. M(, 4); about 7. units. M(6, ); 10 units 3. M ( 7, 1 ); about 9. units 4. x = 3 5. t = 6. p = 3 7. w = 8. x = z = yes; he length can be found using the ythagorean heorem. 1. ( 4, 1) and (0, 7). G(3, 6) and H(9, ) 3. U( 1, ) and V(8, 0) olving quations with Variables on oth ides xample 3 olve 5x = 3x. 5x = 3x Write the equation. +5x +5x dd 5x to each side. = x implify. = x olve the equation. 1 = x implify. he solution is x = 1. ivide each side by. 4. 7x + 1 = 3x t = t 6. 5p + 10 = 8p w + 13 = 11w x + 1 = 3 x 9. z = 4 + 9z 10. ONING Is it possible to find the length of a segment in a coordinate plane without using the istance ormula? xplain your reasoning. ynamic olutions available at igideasmath.com 9 Vocabulary eview Have students make a oncept ircle for each of the following topics. Using the Midpoint ormula to find the midpoint of a segment Using the istance ormula to find the length of a segment olving a linear equation in one variable with the variable on both sides hapter 5 9

5 MONIOING OG NW 1. theorem; It is the lopes of erpendicular Lines heorem (hm. 3.14) studied in ection theorem; It is the Linear air erpendicular heorem (hm. 3.10) studied in ection definition; his is the definition of perpendicular lines. 4. postulate; his is the wo oint ostulate (ost..1) studied in ection.3. In uclidean geometry, it is assumed, not proved, to be true. Mathematical ractices efinitions, ostulates, and heorems ore oncept Mathematically proficient students understand and use given defi nitions. efinitions and iconditional tatements definition is always an if and only if statement. Here is an example. efinition: wo geometric figures are congruent figures if and only if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other. ecause this is a definition, it is a biconditional statement. It implies the following two conditional statements. 1. If two geometric figures are congruent figures, then there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.. If there is a rigid motion or a composition of rigid motions that maps one geometric figure onto another, then the two geometric figures are congruent figures. efinitions, postulates, and theorems are the building blocks of geometry. In two-column proofs, the statements in the reason column are almost always definitions, postulates, or theorems. Monitoring rogress Identifying efinitions, ostulates, and heorems lassify each statement as a definition, a postulate, or a theorem. a. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. b. If two coplanar lines have no point of intersection, then the lines are parallel. c. If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. OLUION a. his is a theorem. It is the lternate Interior ngles onverse heorem (heorem 3.6) studied in ection 3.3. b. his is the definition of parallel lines. c. his is a postulate. It is the arallel ostulate (ostulate 3.1) studied in ection 3.1. In uclidean geometry, it is assumed, not proved, to be true. lassify each statement as a definition, a postulate, or a theorem. xplain your reasoning. 1. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is 1.. If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 3. If two lines intersect to form a right angle, then the lines are perpendicular. 4. hrough any two points, there exists exactly one line. 30 hapter 5 ongruent riangles Laurie s Notes Mathematical ractices (continued from page -9) efinitions, postulates, theorems, and algebraic properties are generally the justification for most of the statements one makes in writing a proof. It is important that students be able to recognize these building blocks of geometry, particularly when constructing an argument or writing a proof. tudents should note that in the example all of the statements are written as if-then (conditional) statements. Give time for students to work through uestions 1 4, and then discuss as a class. 30 hapter 5

6 Laurie s Notes Overview of ection 5.1 Introduction tudents should be familiar with both theorems presented in this lesson. here are many explorations students may have done in middle school to discover that the sum of the interior angles of a triangle is 180 and that the measure of an exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. he approach in this lesson is to reason deductively and write a formal proof of these angle relationships. he relationships are then used in problem solving. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips We are approaching the middle portion of the book and the year. You should be comfortable with a variety of formative assessment techniques, and students should be comfortable with speaking aloud in class and with partners. he goal of your efforts is certainly to improve teaching and learning as result of the activities carried out during the instructional process. lthough formative assessment is often described as a tool to identify certain student conceptions and misconceptions during the instructional phase, it is also a process by which your instruction is improved. ome formative assessment techniques are carried out over a period or unit of instruction. hroughout this chapter, you are encouraged to try a particular technique while still integrating and practicing previously learned techniques. esponse (hinking) Logs: his is a type of writing journal that will be used throughout this chapter after computations, during a conceptual activity, or as part of a problem-solving activity. tudents will respond to a short writing prompt or sentence stem, reflecting on what they are learning. In the response log, students record the process they have gone through to learn something new, any questions they may need to have clarified, what they find challenging, or what information they need to have clarified. he response log allows students to make connections to what they have learned, set goals, and reflect on their learning process. he act of writing helps students become more aware of their own learning and what they can do to self-direct it. he esponse Log is private. he purpose is to promote metacognition. If there are times when you want to collect the response logs, perhaps because you sense a need to redirect your instruction, students should be told this. If there is a possibility that the response log will not remain private, I let my students know this. rovide 4 6 pieces of lined paper stapled together or, alternatively, have students set aside a place in their spiral notebooks. On the left one-third of the paper, the writing stem or prompt is written. he right two-thirds of the paper is where the response is written. I have a large poster in my room with writing stems listed. hroughout the year, students suggest additions to the writing stems. I suggest writing stems throughout, though you should feel free to substitute or leave open for student selection any of the writing stems. or a beginning list of writing stems, see page -38. acing uggestion omplete the explorations and discuss students observations. ransition to the formal lesson. ection

7 ommon ore tate tandards HG-O..10 rove theorems about triangles. HG-MG..1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Laurie s Notes xploration Motivate efore students arrive, cut out large models of triangles. My set of seven triangles is laminated because I use them throughout the year for different discussions. Here s a sample of triangles to cut: sk for seven volunteers. Hand each of them a triangle. Have students sort the triangles by angle measure and then by side length. What do they all have in common? three sides; three angles; he interior angles sum to 180. xplain that in today s lesson the vocabulary of triangles will be reviewed, and students will prove their conjecture about the sum of the interior angles of a triangle. xploration 1 It is quite likely that students are aware that the interior angles of a triangle sum to 180. he focus of the exploration, therefore, is on the process. tudents are using a tool to investigate a relationship and as a result of inductive reasoning a conjecture is made. his process can be contrasted with the deductive approach in the lesson. heck that students are using the dynamic aspect of the software. xploration his exploration, as written, is direct in telling students what to measure. herefore, students do not discover the relationship between an exterior angle and the two nonadjacent interior angles. If you wish students to have the opportunity for this discovery, then change the directions. nother Way: Use dynamic geometry software to draw any and an exterior angle at any vertex. How does the measure of the exterior angle compare to one or more of the interior angles? ample conjectures: he exterior angle is supplementary to the adjacent interior angle. he exterior angle can be acute, right, or obtuse. he exterior angle can be congruent to at most one interior angle. Make your conjecture(s). raw several triangles and test your conjecture(s). egardless of approach, students are using inductive reasoning to make a conjecture. ommunicate Your nswer Have students share their reasoning for uestion 4. onnecting to Next tep he explorations provide an opportunity for students to make conjectures about angle relationships in a triangle. hese conjectures will be proven deductively in the formal lesson. -31 hapter 5

8 5.1 ONUING VIL GUMN o be proficient in math, you need to reason inductively about data and write conjectures. ngles of riangles ssential uestion How are the angle measures of a triangle related? Writing a onjecture Work with a partner. a. Use dynamic geometry software to draw any triangle and label it. b. ind the measures of the interior angles of the triangle. c. ind the sum of the interior angle measures. d. epeat parts (a) (c) with several other triangles. hen write a conjecture about the sum of the measures of the interior angles of a triangle. Writing a onjecture Work with a partner. a. Use dynamic geometry software to draw any triangle and label it. b. raw an exterior angle at any vertex and find its measure. c. ind the measures of the two nonadjacent interior angles of the triangle. d. ind the sum of the measures of the two nonadjacent interior angles. ompare this sum to the measure of the exterior angle. e. epeat parts (a) (d) with several other triangles. hen write a conjecture that compares the measure of an exterior angle with the sum of the measures of the two nonadjacent interior angles. ommunicate Your nswer 3. How are the angle measures of a triangle related? ample ngles m = m = m = n exterior angle of a triangle measures 3. What do you know about the measures of the interior angles? xplain your reasoning. ample ngles m = m = m = ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work. b. heck students work. c. 180 d. heck students work; he sum of the measures of the interior angles of a triangle is a. heck students work. b. heck students work. c. heck students work. d. heck students work; he sum is equal to the measure of the exterior angle. e. heck students work; he measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 3. he sum of the measures of the interior angles of a triangle is 180, and the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 4. he sum of the measures of the two nonadjacent interior angles is 3, and the measure of the adjacent interior angle is 148 ; hese are known because of the conjectures made in xplorations 1 and. ection 5.1 ngles of riangles 31 ection

9 nglish Language Learners lass ctivity Have each student draw a large triangle, use a protractor to measure its angles to the nearest degree, and classify the angles. Use a frequency table to record the number of triangles with 0, 1,, and 3 acute angles. Have students discuss the findings and make conjectures about the number of each type of angle that may be found in any triangle. xtra xample 1 lassify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. 5.1 Lesson What You Will Learn ore Vocabulary interior angles, p. 33 exterior angles, p. 33 corollary to a theorem, p. 35 revious triangle ING Notice that an equilateral triangle is also isosceles. n equiangular triangle is also acute. lassify triangles by sides and angles. ind interior and exterior angle measures of triangles. lassifying riangles by ides and by ngles ecall that a triangle is a polygon with three sides. You can classify triangles by sides and by angles, as shown below. ore oncept lassifying riangles by ides calene riangle Isosceles riangle quilateral riangle no congruent sides at least congruent sides 3 congruent sides lassifying riangles by ngles cute ight Obtuse quiangular riangle riangle riangle riangle 3 acute angles 1 right angle 1 obtuse angle 3 congruent angles lassify the triangular shape of the support beams in the diagram by its sides and by measuring its angles. lassifying riangles by ides and by ngles scalene right triangle MONIOING OG NW 1. ample answer: OLUION he triangle has a pair of congruent sides, so it is isosceles. y measuring, the angles are 55, 55, and 70. o, it is an acute isosceles triangle. Monitoring rogress Help in nglish and panish at igideasmath.com 1. raw an obtuse isosceles triangle and an acute scalene triangle. 3 hapter 5 ongruent riangles Laurie s Notes eacher ctions Whiteboarding: tate the name of each type of triangle, and have students draw a sketch. tudents should mark enough information on the triangle to convey knowledge of the definition of the triangle. robing uestion: Have students sketch triangles that are specified by sides and angles, where not all are possible. xamples: acute isosceles; right scalene; obtuse equilateral (not possible); acute scalene 3 hapter 5

10 lassify O by its sides. hen determine whether it is a right triangle. OLUION lassifying a riangle in the oordinate lane tep 1 Use the istance ormula to find the side lengths. O = (x x 1 ) + (y y 1 ) = ( 1 0) + ( 0) = 5. O = (x x 1 ) + (y y 1 ) = (6 0) + (3 0) = = (x x 1 ) + (y y 1 ) = [6 ( 1)] + (3 ) = ecause no sides are congruent, O is a scalene triangle. tep heck for right angles. he slope of O is is 6 0 = 1 (. he product of the slopes is 1 O is a right angle. o, O is a right scalene triangle. Monitoring rogress ( 1, ) ) =. he slope of O = 1. o, O O and Help in nglish and panish at igideasmath.com. has vertices (0, 0), (3, 3), and ( 3, 3). lassify the triangle by its sides. hen determine whether it is a right triangle. 4 y O(0, 0) (6, 3) x ifferentiated Instruction Kinesthetic Have students draw a large triangle on a sheet of paper and cut it out. hen tear off the corners. rrange the torn pieces so that their vertices coincide and form three adjacent angles. ased on their arrangement, ask students what appears to be true about the sum of the measures of the three interior angles in a triangle. xtra xample lassify by its sides. hen determine whether it is a right triangle. y 4 (0, 3) (7, 1) inding ngle Measures of riangles When the sides of a polygon are extended, other angles are formed. he original angles are the interior angles. he angles that form linear pairs with the interior angles are the exterior angles. (1, 1) 4 6 x is a scalene acute triangle. It is not a right triangle. heorem interior angles heorem 5.1 riangle um heorem he sum of the measures of the interior angles of a triangle is 180. exterior angles MONIOING OG NW. isosceles; right roof p. 34; x. 53, p. 38 m + m + m = 180 ection 5.1 ngles of riangles 33 Laurie s Notes eacher ctions tate the coordinates of O. How can you classify the triangle by its sides and angles? Use the istance ormula to find the length of each side of the triangle. ind the slopes of the sides to see whether two sides are perpendicular. M3 onstruct Viable rguments and ritique the easoning of Others: ome students will try to answer by simply graphing and using eyesight. tudents must be able to provide a viable argument, and eyesight is not sufficient! ommon Misconception: here are two exterior angles at each vertex, not just one. ection

11 xtra xample 3 ind m. (3x + 5) x o prove certain theorems, you may need to add a line, a segment, or a ray to a given diagram. n auxiliary line is used in the proof of the riangle um heorem. riangle um heorem Given rove m 1 + m + m 3 = m = lan for roof a. raw an auxiliary line through that is parallel to 1 3. b. how that m 4 + m + m 5 = 180, 1 4, and 3 5. c. y substitution, m 1 + m + m 3 = 180. lan MN ON in ction a. 1. raw parallel to. 1. arallel ostulate (ost. 3.1) b.. m 4 + m + m 5 = 180. ngle ddition ostulate (ost. 1.4) and definition of straight angle , lternate Interior ngles heorem (hm. 3.) 4. m l = m 4, m 3 = m 5 4. efinition of congruent angles c. 5. m l + m + m 3 = ubstitution roperty of quality heorem heorem 5. xterior ngle heorem he measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. roof x. 4, p m 1 = m + m ind m JKM. OLUION tep 1 Write and solve an equation to find the value of x. inding an ngle Measure (x 5) = 70 + x pply the xterior ngle heorem. x = 75 olve for x. tep ubstitute 75 for x in x 5 to find m JKM. x 5 = 75 5 = 145 o, the measure of JKM is 145. J x 70 (x 5) L K M 34 hapter 5 ongruent riangles Laurie s Notes eacher ctions raw and label 1,, and 3. raw and explain what an auxiliary line is. urn and alk: How could you prove that the three angles sum to 180? Give partners time to talk and plan. Use opsicle ticks to solicit suggestions from students. Write a proof for this theorem. epeat a similar process for the xterior ngle heorem, but do not write a proof if it is part of the homework assignment. esponse Logs: ight now I am thinking about 34 hapter 5

12 corollary to a theorem is a statement that can be proved easily using the theorem. he corollary below follows from the riangle um heorem. orollary orollary 5.1 orollary to the riangle um heorem he acute angles of a right triangle are complementary. roof x. 41, p. 37 Modeling with Mathematics m + m = 90 xtra xample 4 he measure of one acute angle of a right triangle is 1.5 times the measure of the other acute angle. ind the measure of each acute angle. 36 and 54 MONIOING OG NW , 6 x x In the painting, the red triangle is a right triangle. he measure of one acute angle in the triangle is twice the measure of the other. ind the measure of each acute angle. OLUION 1. Understand the roblem You are given a right triangle and the relationship between the two acute angles in the triangle. You need to find the measure of each acute angle.. Make a lan irst, sketch a diagram of the situation. You can use the orollary to the riangle um heorem and the given relationship between the two acute angles to write and solve an equation to find the measure of each acute angle. 3. olve the roblem Let the measure of the smaller acute angle be x. hen the measure of the larger acute angle is x. he orollary to the riangle um heorem states that the acute angles of a right triangle are complementary. Use the corollary to set up and solve an equation. x + x = 90 orollary to the riangle um heorem x = 30 olve for x. o, the measures of the acute angles are 30 and (30 ) = Look ack dd the two angles and check that their sum satisfies the orollary to the riangle um heorem = 90 Monitoring rogress Help in nglish and panish at igideasmath.com 3. ind the measure of ind the measure of each acute angle. 3x x 40 1 (5x 10) (x 6) ection 5.1 ngles of riangles 35 Laurie s Notes eacher ctions act-irst uestioning: he acute angles of a right triangle are complementary. xplain why. Listen for valid reasoning. xplain what a corollary is, and write the orollary to the riangle um heorem. hink-air-hare: Have students answer uestions 3 and 4, and then share and discuss as a class. losure esponse Logs: I figured out that, or What is confusing me the most is. ection

13 ssignment Guide and Homework heck IGNMN asic: 1,, 3 7 odd, 37, 46, 48, verage: 1,, 6 36 even, 37, 38, 4, 43, 46, 48, dvanced: 1,, 10, 1, even, 41 44, 46, HOMWOK HK asic: 3, 11, 17, 1, 5 verage: 8, 18,, 6, 4 dvanced:, 6, 41, 51, xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING an a right triangle also be obtuse? xplain your reasoning.. OML H NN he measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles. Monitoring rogress and Modeling with Mathematics In xercises 3 6, classify the triangle by its sides and by measuring its angles. (ee xample 1.) 3. X Y Z 4. L M N In xercises 15 18, find the measure of the exterior angle. (ee xample 3.) x (x ) 45 NW 1. no; y the orollary to the riangle um heorem (or. 5.1), the acute angles of a right triangle are complementary. ecause their measures have to add up to 90, neither angle could have a measure greater than 90.. nonadjacent 3. right isosceles 4. equiangular equilateral 5. obtuse scalene 6. acute scalene 7. isosceles; right 8. isosceles; not right 9. scalene; not right 10. scalene; right ; acute ; obtuse ; right ; equiangular , 54 0., , , J K 36 hapter 5 ongruent riangles H In xercises 7 10, classify by its sides. hen determine whether it is a right triangle. (ee xample.) 7. (, 3), (6, 3), (, 7) 8. (3, 3), (6, 9), (6, 3) 9. (1, 9), (4, 8), (, 5) (, 3), (0, 3), (3, ) In xercises 11 14, find m 1. hen classify the triangle by its angles (3x + 6) (x + 8) 4 (7x 16) (x + 18) 4x In xercises 19, find the measure of each acute angle. (ee xample 4.) x (6x + 7) (11x ) x 0.. (3x + ) (19x 1) x (13x 5) 36 hapter 5

14 In xercises 3 6, find the measure of each acute angle in the right triangle. (ee xample 4.) 3. he measure of one acute angle is 5 times the measure of the other acute angle. 4. he measure of one acute angle is 8 times the measure of the other acute angle. 5. he measure of one acute angle is 3 times the sum of the measure of the other acute angle and he measure of one acute angle is twice the difference of the measure of the other acute angle and 1. O NLYI In xercises 7 and 8, describe and correct the error in finding m m 1 = m 1 = 360 m 1 = m = 180 m = 180 m 1 = 50 In xercises 9 36, find the measure of the numbered angle UING OOL hree people are standing on a stage. he distances between the three people are shown in the diagram. lassify the triangle by its sides and by measuring its angles. 8 ft 6.5 ft 5 ft 38. UING UU Which of the following sets of angle measures could form a triangle? elect all that apply. 100, 50, 40 96, 74, , 113, 8 101, 41, 38 90, 45, 45 84, 6, MOLING WIH MHMI You are bending a strip of metal into an isosceles triangle for a sculpture. he strip of metal is 0 inches long. he first bend is made 6 inches from one end. escribe two ways you could complete the triangle. 40. HOUGH OVOKING ind and draw an object (or part of an object) that can be modeled by a triangle and an exterior angle. escribe the relationship between the interior angles of the triangle and the exterior angle in terms of the object. 41. OVING OOLLY rove the orollary to the riangle um heorem (orollary 5.1). Given is a right triangle. rove and are complementary. 4. OVING HOM rove the xterior ngle heorem (heorem 5.). Given, exterior rove m + m = m ection 5.1 ngles of riangles 37 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 3. 15, , , , 5 7. he sum of the measures of the angles should be 180 ; m 1 = m 1 = 180 m 1 = 6 8. he measure of the exterior angle should be equal to the sum of the measures of the two nonadjacent interior angles; m 1 = m 1 = acute scalene 38.,,, 39. You could make another bend 6 inches from the first bend and leave the last side 8 inches long, or you could make another bend 7 inches from the first bend and then the last side will also be 7 inches long. 40. ample answer: 1 GO M! 3 When a triangular pennant is on a stick, the two angles on the top edge of the pennant ( 1 and ) should have measures such that their sum is equal to the measure of the angle formed by the stick and the bottom edge of the pennant ( 3) ee dditional nswers. ection

15 NW 43. yes; no n obtuse equilateral triangle is not possible, because when two sides form an obtuse angle the third side that connects them must be longer than the other two. 44. yes; no right equilateral triangle is not possible, because the hypotenuse must be longer than either leg in a right triangle. 45. a. x = 8, x = 9 b. one (x = 4) 46. a. equiangular, acute, equilateral, isosceles b. acute, isosceles c. obtuse, scalene d. right, scalene ee dditional nswers. Mini-ssessment 1. lassify the triangle by its sides and by measuring its angles. obtuse isosceles triangle. ind the measure of the exterior angle. 43. IIL HINKING Is it possible to draw an obtuse isosceles triangle? obtuse equilateral triangle? If so, provide examples. If not, explain why it is not possible. 44. IIL HINKING Is it possible to draw a right isosceles triangle? right equilateral triangle? If so, provide an example. If not, explain why it is not possible. 45. MHMIL ONNION is isosceles, = x, and = x 4. a. ind two possible values for x when the perimeter of is 3. b. How many possible values are there for x when the perimeter of is 1? 46. HOW O YOU I? lassify the triangles, in as many ways as possible, without finding any measurements. a. b. c. d. 47. NLYZING LIONHI Which of the following could represent the measures of an exterior angle and two interior angles of a triangle? elect all that apply. 100, 6, 38 81, 57, 4 119, 68, 49 95, 85, 8 9, 78, , 101, 48 Maintaining Mathematical roficiency 48. MKING N GUMN Your friend claims the measure of an exterior angle will always be greater than the sum of the nonadjacent interior angle measures. Is your friend correct? xplain your reasoning. MHMIL ONNION In xercises 49 5, find the values of x and y y y 75 x x x y 0 5. y x OVING HOM Use the diagram to write a proof of the riangle um heorem (heorem 5.1). Your proof should be different from the proof of the riangle um heorem shown in this lesson. Use the diagram to find the measure of the segment or angle. (ection 1. and ection 1.5) 54. m KHL L 55. m G 5y 8 K 3y 8z GH (6x + ) (5x 7) (3x + 1) 57. 3z + 6 H eviewing what you learned in previous grades and lessons (x + 11) 38 hapter 5 ongruent riangles x he measure of one acute angle of a right triangle is 6 less than twice the measure of the other acute angle. ind the measure of each acute angle. 3 and has vertices ( 6, ), (5, 1), and (4, 4). lassify by its sides. hen determine whether it is a right triangle. scalene triangle; yes, it is a right triangle. If students need help... esources by hapter ractice and ractice uzzle ime tudent Journal ractice ifferentiating the Lesson kills eview Handbook If students got it... esources by hapter nrichment and xtension umulative eview tart the next ection 38 hapter 5

16 Laurie s Notes Overview of ection 5. Introduction tudents have previously studied congruent angles and congruent segments. In this lesson, congruency is extended to polygons. tudents will identify and use corresponding parts of congruent polygons to solve problems. rigid motion is an isometry, so segment lengths and angle measures are preserved. his allows us to say that the corresponding parts of congruent figures are congruent. he properties of congruencereflexive, symmetric, and transitiveare stated for triangles. he hird ngles heorem (hm. 5.4) is presented, with the proof left for the exercises. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations esources racing paper and dynamic geometry software are used in the explorations. ommon Misconceptions When naming congruent polygons that have more than three sides, you need to pay attention to how the vertices are listed. Given WXYZ, it is not true that WYZX even though the same letters are still matched in each statement. he vertices must be in order when naming the polygon. is a diagonal, not a side. ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. eginning list of writing stems: sticky part for me is t first I thought but now I think I am feeling good about I didn t expect I feel confident about I feel like I made progress today I figured out that I figured these out because I got stuck I learned to I made progress with I need to rethink I was really surprised when I was successful in I will understand this better if I I wish I could I wish I had I m not sure I m still not sure about It helped me today when ight now I am thinking about ight now I know ight now I need to he hardest part right now is omorrow I need to find out What confused me the most was What is confusing me the most is What is puzzling me the most is acing uggestion omplete the explorations and then start the formal lesson with a discussion of corresponding parts of congruent figures. ection

17 ommon ore tate tandards HG-O..7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Laurie s Notes xploration Motivate tory ime: ell students about an antique patchwork quilt that you purchased recently that needs a bit of repair. here are several triangular pieces, each different from the others, that are worn and need to be replaced. What would a quilt restorer need to measure in order to repair the quilt? tudents should recognize that there are three sides and three angles for each triangle and these six parts would need to be measured. xploration Note o show that two triangles are congruent using a transformational approach, you need to perform one or more rigid motions to map one triangle to the other. In xploration, students explore what this would look like for a triangle that begins in uadrant II and, after a composition of rigid motions, has its image in uadrant IV. xploration 1 his first exploration should be a fairly quick discussion between partners. Have students share their thinking. xploration his exploration can be challenging for students with weaker spatial skills. eaching ip: Use tracing paper and draw. tudents can manipulate the tracing paper to make a conjecture about the composition of rigid motions needed to map to. irculate to see that students are making a conjecture about the rigid motions that map to versus using trial and error with the dynamic software. What do you know about and? xplain. hey are congruent. Listen for valid reasoning. xtension: Have students explore the orientation, clockwise (W) or counter-clockwise (W), of the letters labeling the vertices of the triangle. reflection will change the orientation of the letters. or the original figure,, the orientation of the letters is W. If the orientation of the letters in is W, then an odd number of reflections have been performed. If the orientation of the letters in is W, then no reflection has occurred or an even number of reflections have been performed. ommunicate Your nswer Listen to student discussion. composition of isometries results in the image being congruent to the original figure. onnecting to Next tep he explorations use rigid motions taught in the previous chapter to introduce corresponding parts of congruent figures, which are presented in the formal lesson. -39 hapter 5

18 LOOKING O UU 5. o be proficient in math, you need to look closely to discern a pattern or structure. ongruent olygons ssential uestion Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? escribing igid Motions Work with a partner. Of the four transformations you studied in hapter 4, which are rigid motions? Under a rigid motion, why is the image of a triangle always congruent to the original triangle? xplain your reasoning. ranslation eflection otation ilation inding a omposition of igid Motions Work with a partner. escribe a composition of rigid motions that maps to. Use dynamic geometry software to verify your answer. a. b c. d ommunicate Your nswer Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? 4. he vertices of are (1, 1), (3, ), and (4, 4). he vertices of are (, 1), (0, 0), and ( 1, ). escribe a composition of rigid motions that maps to ection 5. ongruent olygons ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. translation, reflection, rotation; rigid motion maps each part of a figure to a corresponding part of its image. ecause rigid motions preserve length and angle measure, corresponding parts of a figure and its image are congruent. In triangles, this means that the corresponding sides and corresponding angles are congruent, which is sufficient to say that the triangles are congruent.. a. ample answer: a translation 3 units right followed by a reflection in the x-axis b. ample answer: a 180 rotation about the origin c. ample answer: a 70 counterclockwise rotation about the origin followed by a translation 3 units down d. ample answer: a 70 clockwise rotation about the origin followed by a reflection in the y-axis 3. Look at the orientation of the original triangle and decide which rigid motion or composition of rigid motions will result in the same orientation as the second triangle. hen, if necessary, use a translation to move the first triangle so that it coincides with the second. 4. ample answer: a reflection in the y-axis followed by a translation 3 units right and units down ection 5. 39

19 nglish Language Learners Visual id reate a poster to display corresponding parts of two congruent triangles and several true congruence statements using various orders for the vertices of the triangles. Use color to indicate corresponding parts of the triangles and congruence statements. xtra xample 1 Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. N M Z Y MN YXZ; N X, Z, M Y, MN YX, N XZ, M ZY X 5. Lesson What You Will Learn ore Vocabulary corresponding parts, p. 40 revious congruent figures UY I Notice that both of the following statements are true. 1. If two triangles are congruent, then all their corresponding parts are congruent.. If all the corresponding parts of two triangles are congruent, then the triangles are congruent. VIUL ONING o help you identify corresponding parts, rotate. Identify and use corresponding parts. Use the hird ngles heorem. Identifying and Using orresponding arts ecall that two geometric figures are congruent if and only if a rigid motion or a composition of rigid motions maps one of the figures onto the other. rigid motion maps each part of a figure to a corresponding part of its image. ecause rigid motions preserve length and angle measure, corresponding parts of congruent figures are congruent. In congruent polygons, this means that the corresponding sides and the corresponding angles are congruent. When is the image of after a rigid motion or a composition of rigid motions, you can write congruence statements for the corresponding angles and corresponding sides. orresponding angles orresponding sides,,,, When you write a congruence statement for two polygons, always list the corresponding vertices in the same order. You can write congruence statements in more than one way. wo possible congruence statements for the triangles above are or. When all the corresponding parts of two triangles are congruent, you can show that the triangles are congruent. Using the triangles above, first translate so that point maps to point. his translation maps to. Next, rotate counterclockwise through so that the image of coincides with. ecause, the rotation maps point to point. o, this rotation maps to. Now, reflect in the line through points and. his reflection maps the sides and angles of to the corresponding sides and corresponding angles of, so. o, to show that two triangles are congruent, it is sufficient to show that their corresponding parts are congruent. In general, this is true for all polygons. J Identifying orresponding arts L K Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. OLUION he diagram indicates that JKL. orresponding angles J, K, L orresponding sides JK, KL, LJ J L K 40 hapter 5 ongruent riangles Laurie s Notes eacher ctions robing uestions: What does it mean for two objects to be congruent? nswers will vary. What does it mean for two polygons to be congruent? nswers will vary. raw and label two congruent triangles. What does it mean for two triangles to be congruent? tudents should make a list (i.e., corresponding parts) of what would be true. M3 onstruct Viable rguments and ritique the easoning of Others and urn and alk: How could rigid motions be used to show that two triangles are congruent? Listen for valid reasoning. iscuss notation and marking diagrams of congruent polygons. 40 hapter 5

20 In the diagram, G. a. ind the value of x. b. ind the value of y. OLUION a. You know that G. Using roperties of ongruent igures b. You know that. G = m = m 1 = x 4 68 = (6y + x) 16 = x 68 = 6y = x 10 = y You divide the wall into orange and blue sections along JK. Will the sections of the wall be the same size and shape? xplain. 8 ft (x 4) ft 10 (6y + x) G 1 ft howing hat igures re ongruent J 1 xtra xample In the diagram, G MN. N 8 ft 10 (3x + y) 84 G 68 1 ft a. ind the value of x. 10 b. ind the value of y. 7 M(x ) ft xtra xample 3 how that. xplain your reasoning. OLUION rom the diagram, and 3 4 because all right angles are congruent. lso, K by the Lines erpendicular to a ransversal heorem (hm. 3.1),. hen 1 4 and 3 by the lternate Interior ngles heorem (hm. 3.). o, all pairs of corresponding angles are congruent. he diagram shows J K, K J, and. y the eflexive roperty of ongruence (hm..1), JK KJ. o, all pairs of corresponding sides are congruent. ecause all corresponding parts are congruent, JK KJ. Yes, the two sections will be the same size and shape. Monitoring rogress In the diagram, GH. 1. Identify all pairs of congruent corresponding parts.. ind the value of x. (4x + 5) H G 3. In the diagram at the left, show that. Help in nglish and panish at igideasmath.com rom the diagram, and. y the eflexive roperty of ongruence (hm..1),. o, all pairs of corresponding sides are congruent. rom the diagram, and. ecause, by the lternate Interior ngles heorem (hm. 3.). o, all pairs of corresponding angles are congruent. ecause all corresponding parts are congruent,. UY I he properties of congruence that are true for segments and angles are also true for triangles. Laurie s Notes heorem heorem 5.3 roperties of riangle ongruence riangle congruence is reflexive, symmetric, and transitive. eflexive or any triangle,. ymmetric If, then. ransitive If and JKL, then JKL. roof igideasmath.com eacher ctions ection 5. ongruent olygons 41 xtension: Is there another way to write the congruence statement for G? xplain. yes; G, as long as the corresponding vertices are in the same order. act-irst uestioning: he eflexive, ymmetric, and ransitive roperties can be applied to triangle congruence. xplain what this means and why it is true. Listen for correct statements and valid reasoning. Write the roperties of riangle ongruence. MONIOING OG NW 1. corresponding angles:,, G, H ; corresponding sides:, G, GH, H. x = 5 3. rom the diagram,,, and. lso, by the Vertical ngles ongruence heorem (hm..6),. om the diagram,, and and by the lternate Interior ngles heorem (hm. 3.). ecause all corresponding parts are congruent,. ection 5. 41

21 xtra xample 4 ind m Using the hird ngles heorem heorem heorem 5.4 hird ngles heorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. roof x. 19, p. 44 If and, then. xtra xample 5 Use the information in the figure to prove that WXY ZVY. W X Y V 45 N 30 ind m. OLUION Using the hird ngles heorem and, so by the hird ngles heorem,. y the riangle um heorem (heorem 5.1), m = = 105. o, m = m = 105 by the definition of congruent angles. roving hat riangles re ongruent tatements (easons) 1. WX ZV, XY VY, WY ZY (Given). X and V are right angles. (Given) 3. X V (ll rt. s.) 4. WYX VYZ (Vertical s hm..6) 5. W Z (hird ngles hm. 5.4) 6. WXY ZVY (ll corr. parts.) MONIOING OG NW , N N Z Use the information in the figure to prove that. OLUION Given,,, rove lan for roof lan in ction a. Use the eflexive roperty of ongruence (hm..1) to show that. b. Use the hird ngles heorem to show that. MN 1., a.. 3., ON 1. Given. eflexive roperty of ongruence (heorem.1) 3. Given b hird ngles heorem ll corresponding parts are congruent. Monitoring rogress Help in nglish and panish at igideasmath.com N Use the diagram. 4. ind m N. 5. What additional information is needed to conclude that N N? 4 hapter 5 ongruent riangles Laurie s Notes eacher ctions M3: In xample 4, students may know that m = 105, but can they explain why? o not ignore the reasoning behind the answer. In xample 5, draw the diagram and pose the question. urn and alk: How could you prove that the two triangles are congruent? Give partners time to talk and plan. Use opsicle ticks to solicit suggestions from students. sk a volunteer to write the proof. losure esponse Logs: elect from What is puzzling me the most is or I was successful in or I m not sure. 4 hapter 5

22 5. xercises Vocabulary and ore oncept heck 1. WIING ased on this lesson, what information do you need to prove that two triangles are congruent? xplain your reasoning.. IN WO, M UION Which is different? ind both answers. Monitoring rogress and Modeling with Mathematics In xercises 3 and 4, identify all pairs of congruent corresponding parts. hen write another congruence statement for the polygons. (ee xample 1.) GHJK H G J K In xercises 5 8, XYZ MNL. opy and complete the statement. 5. m Y = X 6. m M = 7. m Z = 8. XY = Is JKL? Is JLK? 33 L Y Z M 1 N 14 8 In xercises 9 and 10, find the values of x and y. (ee xample.) 9. GH 135 (4y 4) H G 8 Is KJL? Is LKJ? (10x + 65) ynamic olutions available at igideasmath.com J 10. MN U N 14 (x y) m M 4 ection 5. ongruent olygons m (x 50) In xercises 11 and 1, show that the polygons are congruent. xplain your reasoning. (ee xample 3.) 11. W V Z X Y J N 1. X Y W Z In xercises 13 and 14, find m 1. (ee xample 4.) 13. L M Y 70 1 N X Z 14. K K M L L U ssignment Guide and Homework heck IGNMN asic: 1,, 3 17 odd,, 6 9 verage: 1,, 4, even, 1 3, 6 9 dvanced: 1,, even, 0 9 HOMWOK HK asic: 3, 9, 11, 13, 15 verage: 4, 10, 14, 16, 3 dvanced: 10, 14, 16,, 4 NW 1. o show that two triangles are congruent, you need to show that all corresponding parts are congruent. If two triangles have the same side lengths and angle measures, then they must be the same size and shape.. Is JLK?; no; yes 3. corresponding angles:,, ; corresponding sides:,, ; ample answer: 4. corresponding angles: G, H, J, K ; corresponding sides: GH, HJ, JK, GK ; ample answer: HJKG x = 7, y = x = 3; y = rom the diagram, WX LM, XY MN, YZ NJ, VZ KJ, and WV LK. lso from the diagram, V K, W L, X M, Y N, and Z J. ecause all corresponding parts are congruent, VWXYZ KLMNJ. 1. rom the diagram, WX YZ and XY ZW. y the eflexive roperty of ongruence (hm..1), WY WY. lso from the diagram, X Z. hen, from the markings in the diagram, XY ZW and XW ZY. You can conclude that XYW ZWY and XWY ZYW by the lternate Interior ngles heorem (hm. 3.). ecause all corresponding parts are congruent, WXY YZW ection 5. 43

23 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations 15. OO riangular postage stamps, like the ones shown, are highly valued by stamp collectors. rove that. (ee xample 5.) 19. OVING HOM rove the hird ngles heorem (heorem 5.4) by using the riangle um heorem (heorem 5.1). 0. HOUGH OVOKING raw a triangle. opy the triangle multiple times to create a rug design made of congruent triangles. Which property guarantees that all the triangles are congruent? NW ee dditional nswers. 17. he congruence statement should be used to ensure that corresponding parts are matched up correctly; Y; m = m Y; m = 90 4 = In order to conclude that triangles are congruent, the sides must also be congruent; MN is not congruent to because the corresponding sides are not congruent ee dditional nswers. Mini-ssessment 1.. Name all pairs of congruent corresponding parts. hen write another congruence statement for the triangles.,,,,, ; ample answer:. uadrilateral GH quadrilateral. ind the value of x. 99 Given,, is the midpoint of and. rove 16. OO Use the information in the figure to prove that G. O NLYI In xercises 17 and 18, describe and correct the error. 17. Given XZY Y Z 4 m = m Z X Z m = M 44 hapter 5 ongruent riangles G N MN because the corresponding angles are congruent. Maintaining Mathematical roficiency What can you conclude from the diagram? (ection 1.6) 6. X V 7. N 8. Y Z U W J 1. ONING JKL is congruent to XYZ. Identify all pairs of congruent corresponding parts.. HOW O YOU I? In the diagram,. a. xplain how you know that and. b. xplain how you know that G G. c. xplain how you know that G G. d. o you have enough information to prove that G G? xplain. MHMIL ONNION In xercises 3 and 4, use the given information to write and solve a system of linear equations to find the values of x and y. 3. LMN, m L = 40, m M = 90, m = (17x y), m = (x + 4y) 4. U XYZ, m = 8, m U = (4x + y), m X = 130, m Y = (8x 6y) 5. OO rove that the criteria for congruent triangles in this lesson is equivalent to the definition of congruence in terms of rigid motions. eviewing what you learned in previous grades and lessons L 9. K M G G I H H (6x + 3) G 81 x = 16 If students need help... If students got it What additional information is needed to prove that U? esources by hapter ractice and ractice uzzle ime esources by hapter nrichment and xtension umulative eview tudent Journal ractice tart the next ection U, U U ifferentiating the Lesson kills eview Handbook 44 hapter 5

24 Laurie s Notes Overview of ection 5.3 Introduction he ide-ngle-ide () ongruence heorem, the first of the triangle congruence theorems, is presented in this lesson. It is first investigated in the exploration and then proven in the formal lesson. ifferent approaches are used in the proof examples in the formal lesson: transformational, two-column, and paragraph. One additional construction is presentedhow to copy a triangle using the ongruence heorem. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations esources ynamic geometry software is used in the exploration, and a compass and straightedge are used in the lesson. ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. nother Way he classic, traditional approach to introducing the ongruence heorem is to use objects such as coffee stirrers to represent two sides of a triangle and then fix an angle measure between them. egardless of the orientation of the stirrers, the triangles formed will all be congruent. acing uggestion ake time for students to explore and discover the congruence relationship using dynamic geometry software, and then transition to the formal lesson. ection

25 ommon ore tate tandards HG-O..8 xplain how the criteria for triangle congruence (, ) follow from the definition of congruence in terms of rigid motions. HG-MG..1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Laurie s Notes xploration Motivate ose the following problem to get students thinking about triangles. raw six segments of equal length to form a closed figure that contains eight equilateral triangles. Give students time to fool around with the problem. Here is one solution. he figure has six small equilateral triangles and two large equilateral triangles, for a total of eight equilateral triangles. xploration he last lesson established that to show two triangles are congruent it is necessary to show that all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. he ide-ngle-ide () ongruence heorem (hm. 5.5) is the first of several theorems to show that less than six pieces of information are needed to prove that two triangles are congruent. his exploration may not be a familiar approach to verifying the triangle congruence method, so become familiar with the construction steps and potential for extensions. xploration 1 M5 Use ppropriate ools trategically: tudents should be familiar enough with the software to follow the construction described. he software will label points in the construction process that may differ from how the diagram and table are labeled. ell students to make adjustments as needed. he ordered pairs for points and will differ for each student, but the length of the third side and the two missing angles will be the same. In constructing, students should try each orientation, clockwise and counterclockwise. re the results the same? Have students discuss the results of their explorations and make conjectures. xtension: epeat the construction for two circles of different radii or for a different angle. ommunicate Your nswer M3 onstruct Viable rguments and ritique the easoning of Others: tudents should conclude that when two pairs of sides and the included angle of two triangles are congruent, the triangles will be congruent. onnecting to Next tep he exploration allows students to discover and make sense of the congruence relationship that will be presented formally in the lesson. -45 hapter 5

26 UING OOL GILLY 5.3 o be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data. roving riangle ongruence by ssential uestion What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? rawing riangles Work with a partner. Use dynamic geometry software. a. onstruct circles with radii of units and 3 units centered at the origin. onstruct a 40 angle with its vertex at the origin. Label the vertex. b. Locate the point where one ray of the angle intersects the smaller circle and label this point. Locate the point where the other ray of the angle intersects the larger circle and label this point. hen draw. c. ind, m, and m. d. epeat parts (a) (c) several times, redrawing the angle in different positions. Keep track of your results by copying and completing the table below. What can you conclude? m m m 1. (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work b. heck students work. c. 1.95, m 98.79, m 41.1 d. heck students work; If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.. he triangles are congruent. 3. tart with two triangles so that two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. hen show that one triangle can be translated until it coincides with the other triangle by a composition of rigid motions. ommunicate Your nswer. What can you conclude about two triangles when you know that two pairs of corresponding sides and the corresponding included angles are congruent? 3. How would you prove your conclusion in xploration 1(d)? ection 5.3 roving riangle ongruence by 45 ection

27 ifferentiated Instruction uditory/visual oint out that to use the ongruence heorem (hm. 5.5), the congruent angles must be included between the pairs of corresponding congruent sides. he abbreviation illustrates that fact by placing the letter between the two s. 5.3 Lesson ore Vocabulary revious congruent figures rigid motion UY I he included angle of two sides of a triangle is the angle formed by the two sides. What You Will Learn Use the ide-ngle-ide () ongruence heorem. olve real-life problems. Using the ide-ngle-ide ongruence heorem heorem heorem 5.5 ide-ngle-ide () ongruence heorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 46 ide-ngle-ide () ongruence heorem Given,, rove irst, translate so that point maps to point, as shown below. 46 hapter 5 ongruent riangles his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below. ecause, the rotation maps point to point. o, this rotation maps to. Now, reflect in the line through points and, as shown below. ecause points and lie on, this reflection maps them onto themselves. ecause a reflection preserves angle measure and, the reflection maps to. ecause, the reflection maps point to point. o, this reflection maps to. ecause you can map to using a composition of rigid motions,. Laurie s Notes eacher ctions he ide-ngle-ide () ongruence heorem should make sense following the exploration. raw the diagram of the two triangles as shown. Mark the triangles. urn and alk: How could you use a composition of rigid motions to map to? fter partners have discussed, solicit ideas. tudents will likely suggest a translation, rotation, and then reflection. M5 and eaching ip: onstruct a similar image using construction software. erform the steps in the proof as students suggest them. 46 hapter 5

28 UY I Make your proof easier to read by identifying the steps where you show congruent sides () and angles (). Write a proof. Given, rove OLUION MN 1.. Using the ongruence heorem ON 1. Given. Given lternate Interior ngles heorem (hm. 3.) eflexive roperty of ongruence (hm..1) ongruence heorem Using and roperties of hapes In the diagram, and pass through the center M of the circle. What can you conclude about M and M? M OLUION ecause they are vertical angles, M M. ll points on a circle are the same distance from the center, so M, M, M, and M are all congruent. o, M and M are congruent by the ongruence heorem. Monitoring rogress Help in nglish and panish at igideasmath.com In the diagram, is a square with four congruent sides and four right angles.,,, and U are the midpoints of the sides of. lso, U and V VU. xtra xample 1 Write a proof. Given is the midpoint of. and are right angles. rove tatements (easons) 1. is the midpoint of. (Given). (ef. of midpoint) 3. and are right angles. (Given) 4. (ight s hm..3) 5. (eflex. rop. hm..1) 6. ( hm. 5.5) xtra xample What can you conclude about and? 1. rove that V UV.. rove that U. V U ecause they are vertical angles,. o, by the ongruence heorem (hm. 5.5). ection 5.3 roving riangle ongruence by 47 MONIOING OG NW 1. ee dditional nswers. Laurie s Notes eacher ctions ose xample 1. Have students mark their diagrams with known information or information that can be concluded from given information. Work through the proof as shown. ompare and contrast this synthetic, two-column proof with the transformational proof on page 46. M3: he solution in xample is a paragraph style proof. Valid reasoning outlines how to show that the triangles are congruent. esponse Logs: elect from I made progress with or I got stuck. ection

29 xtra xample 3 he wings of a paper airplane have two congruent sides and two congruent angles. Use the ongruence heorem (hm. 5.5) to show that I I. I opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OLUION tep 1 tep tep 3 tep 4 You are given that I I and I I. y the eflexive roperty of ongruence (hm..1), I I. o, two pairs of sides and their included angles are congruent. herefore, I I by the heorem (hm. 5.5). onstruct a side onstruct so that it is congruent to. onstruct an angle onstruct with vertex and side so that it is congruent to. onstruct a side onstruct so that it is congruent to. olving eal-life roblems olving a eal-life roblem raw a triangle raw. y the ongruence heorem,. MONIOING OG NW 3. ee dditional nswers. You are making a canvas sign to hang on the triangular portion of the barn wall shown in the picture. You think you can use two identical triangular sheets of canvas. You know that and. Use the ongruence heorem to show that. OLUION You are given that. y the eflexive roperty of ongruence (heorem.1),. y the definition of perpendicular lines, both and are right angles, so they are congruent. o, two pairs of sides and their included angles are congruent. and are congruent by the ongruence heorem. Monitoring rogress Help in nglish and panish at igideasmath.com 3. You are designing the window shown in the photo. You want to make congruent to G. You design the window so that G and G. Use the ongruence heorem to prove G. G 48 hapter 5 ongruent riangles Laurie s Notes hs_geo_pe_0503.indd 48 eacher ctions Have students draw different types of triangles and label them. his construction involves copying an angle and copying two sides. Have students compare their work with one another. eaching ip: In xample 3, draw with perpendicular bisector without the photo behind it. Mark the diagram. losure esponse Logs: elect from I feel confident about or t first I thought but now I think. 1/19/15 10:8 M 48 hapter 5

30 5.3 xercises Vocabulary and ore oncept heck In xercises 3 8, name the included angle between the pair of sides given. K J 3. JK and KL 4. K and LK 5. L and LK 6. JL and JK 7. KL and JL 8. K and L In xercises 9 14, decide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (heorem 5.5). xplain. 9., 10. LMN, N 11. YXZ, WXZ 1. V, U Z W X Y L L N M V U 13. H, GH 14. KLM, MNK G 1. WIING What is an included angle? H N K M ynamic olutions available at igideasmath.com. OML H NN If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then. Monitoring rogress and Modeling with Mathematics L In xercises 15 18, write a proof. (ee xample 1.) 15. Given bisects, rove ection 5.3 roving riangle ongruence by Given, rove 17. Given is the midpoint of and. rove 18. Given, 1 rove ssignment Guide and Homework heck IGNMN asic: 1,, 3 15 odd, 19, 5, 6, 30, 3 35 verage: 1,, 8 4 even, 5, 6, 9, 30, 3 35 dvanced: 1,, 10 4 even, 5 35 HOMWOK HK asic: 3, 9, 11, 15, 19 verage: 10, 16,, 4, 6 dvanced: 14, 16,, 6, 9 NW 1. an angle formed by two sides. the two triangles are congruent 3. JKL 4. KL 5. KL 6. KJL 7. JLK 8. KL 9. no; he congruent angles are not the included angles. 10. yes; wo pairs of sides and the included angles are congruent. 11. no; One of the congruent angles is not the included angle. 1. no; he congruent angles are not the included angles. 13. yes; wo pairs of sides and the included angles are congruent. 14. no; NKM and KML are congruent by the lternate Interior ngles heorem (hm. 3.) but they are not the included angles ee dditional nswers. ection

31 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW ee dditional nswers. Mini-ssessment 1. Use the ongruence heorem to show that WXZ YZX. X W Y Z You are given XW ZY and WXZ YZX. y the eflexive roperty of ongruence (hm..1), XZ ZX. o, WXZ YZX by the ongruence heorem (hm. 5.5).. he two triangles in the diagram represent congruent banners. You know that and. What pair of angles must be congruent to show that by the ongruence heorem? In xercises 19, use the given information to name two triangles that are congruent. xplain your reasoning. (ee xample.) 19. U, and 0. is a square with is the center of four congruent sides and the circle. four congruent angles. U 1. UV is a regular. MK MN, KL NL, pentagon. and M and L are centers of circles. K U 10 m M L V 10 m N ONUION In xercises 3 and 4, construct a triangle that is congruent to using the ongruence heorem (heorem 5.5) O NLYI escribe and correct the error in finding the value of x. 4x + 6 X Y 5x 5 Z 5x 1 W 3x + 9 4x + 6 = 3x + 9 x + 6 = 9 x = 3 Maintaining Mathematical roficiency 6. HOW O YOU I? What additional information do you need to prove that? lassify the triangle by its sides and by measuring its angles. (ection 5.1) OO he Navajo rug is made of isosceles triangles. You know. Use the ongruence heorem (heorem 5.5) to show that. (ee xample 3.) 8. HOUGH OVOKING here are six possible subsets of three sides or angles of a triangle:,,,,, and. Which of these correspond to congruence theorems? or those that do not, give a counterexample. 9. MHMIL ONNION rove that. hen find the values 4y 6 of x and y. x + 6 3y + 1 4x 30. MKING N GUMN Your friend claims it is possible to construct a triangle congruent to by first constructing and, and then copying. Is your friend correct? xplain your reasoning. 31. OVING HOM rove the eflections in Intersecting Lines heorem (heorem 4.3). eviewing what you learned in previous grades and lessons 3. Write a proof. Given bisects, rove 50 hapter 5 ongruent riangles If students need help... esources by hapter ractice and ractice uzzle ime If students got it... esources by hapter nrichment and xtension umulative eview tudent Journal ractice tart the next ection ifferentiating the Lesson kills eview Handbook ee dditional nswers. 50 hapter 5

32 Laurie s Notes Overview of ection 5.4 Introduction In this lesson, there are two theorems, two corollaries, and a construction. he two theorems, investigated in the exploration, are converses of each other. tudents think of an isosceles triangle as being symmetric, so having base angles congruent is not a surprise. he challenge for students in proving the theorem is that an auxiliary line is drawn. he two corollaries are converses of each other, and when combined as a biconditional, the statement says a triangle is equilateral if and only if it is equiangular. tudents will construct an equilateral triangle. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. acing uggestion Once students have discovered the relationship about the angles in an isosceles triangle using dynamic geometry software, transition to the formal lesson. ection

33 ommon ore tate tandards HG-O..10 rove theorems about triangles. HG-O..13 onstruct an equilateral triangle, HG-MG..1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Laurie s Notes xploration Motivate Isosceles and equilateral triangles appear in many figures that students will study in art and architecture. ell students that in this lesson they will learn properties of triangles that have two or more congruent sides or angles. xploration 1 M5 Use ppropriate ools trategically: tudents should be familiar enough with the software to follow the construction described. he software will label points in the construction process that may differ from how the diagram and table are labeled. ell students to make adjustments as needed. he ordered pairs for points and will differ for each student, as will the length of. Have students discuss the results of their explorations and make conjectures. How do you classify by its sides? It is isosceles. ould it be equilateral? xplain. yes. It is equilateral when = 3. How do you classify by its angles? It varies. It could be acute, right, or obtuse. M3 onstruct Viable rguments and ritique the easoning of Others and M6 ttend to recision: What conjecture can you write about the angle measures of an isosceles triangle? e sure that students refer to the angles opposite the two congruent sides as being congruent. It is not enough to say that two of the three angles in an isosceles triangle are congruent. iscuss the converse of their conjecture. ommunicate Your nswer M3: Listen for valid student reasoning as they describe how they could prove their conjectures. onnecting to Next tep Now that students have investigated a property of the base angles of an isosceles triangle, they should be prepared to write a proof in the formal lesson. -51 hapter 5

34 5.4 ONUING VIL GUMN o be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth of your conjectures. ample quilateral and Isosceles riangles ssential uestion What conjectures can you make about the side lengths and angle measures of an isosceles triangle? Writing a onjecture about Isosceles riangles Work with a partner. Use dynamic geometry software. a. onstruct a circle with a radius of 3 units centered at the origin. b. onstruct so that and are on the circle and is at the origin ample oints (0, 0) (.64, 1.4) ( 1.4,.64) egments = 3 = 3 = 4.4 ngles m = 90 m = 45 m = 45 c. ecall that a triangle is isosceles if it has at least two congruent sides. xplain why is an isosceles triangle. d. What do you observe about the angles of? e. epeat parts (a) (d) with several other isosceles triangles using circles of different radii. Keep track of your observations by copying and completing the table below. hen write a conjecture about the angle measures of an isosceles triangle. m m m 1. (0, 0) (.64, 1.4) ( 1.4,.64) (0, 0) 3. (0, 0) 4. (0, 0) 5. (0, 0) f. Write the converse of the conjecture you wrote in part (e). Is the converse true? ommunicate Your nswer. What conjectures can you make about the side lengths and angle measures of an isosceles triangle? 3. How would you prove your conclusion in xploration 1(e)? in xploration 1(f)? ection 5.4 quilateral and Isosceles riangles 51 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work. b. heck students work. c. ecause all points on a circle are the same distance from the center,. d. e. heck students work; If two sides of a triangle are congruent, then the angles opposite them are congruent. f. If two angles of a triangle are congruent, then the sides opposite them are congruent; yes. In an isosceles triangle, two sides are congruent, and the angles opposite them are congruent. 3. raw the angle bisector of the included angle between the congruent sides to divide the given isosceles triangle into two triangles. Use the ongruence heorem (hm. 5.5) to show that these two triangles are congruent. hen, use properties of congruent triangles to show that the two angles opposite the shared sides are congruent. or the converse, draw the angle bisector of the angle that is not congruent to the other two. his divides the given triangle into two triangles that have two pairs of corresponding congruent angles. he third pair of angles are congruent by the hird ngles heorem (hm. 5.4). lso, the angle bisector is congruent to itself by the eflexive roperty of ongruence (hm..1). o, the triangles are congruent, and the sides opposite the congruent angles in the original triangle are congruent. ection

35 ifferentiated Instruction Kinesthetic Have each student draw and cut out a large isosceles triangle and label the points. Have students fold the triangle so that the congruent sides coincide, and then identify the base, legs, base angles, and vertex angle. 5.4 Lesson ore Vocabulary legs, p. 5 vertex angle, p. 5 base, p. 5 base angles, p. 5 What You Will Learn Use the ase ngles heorem. Use isosceles and equilateral triangles. Using the ase ngles heorem triangle is isosceles when it has at least two congruent sides. When an isosceles triangle has exactly two congruent sides, these two sides are the legs. he angle formed by the legs is the vertex angle. he third side is the base of the isosceles triangle. he two angles adjacent to the base are called base angles. heorems heorem 5.6 ase ngles heorem If two sides of a triangle are congruent, then the angles opposite them are congruent. If, then. vertex angle leg base angles base leg roof p. 5 heorem 5.7 onverse of the ase ngles heorem If two angles of a triangle are congruent, then the sides opposite them are congruent. If, then. roof x. 7, p. 75 ase ngles heorem Given rove lan for roof a. raw so that it bisects. b. Use the ongruence heorem to show that. c. Use properties of congruent triangles to show that. lan MN in ction a. 1. raw, the angle bisector of. ON 1. onstruction of angle bisector.. efinition of angle bisector Given eflexive roperty of ongruence (hm..1) b ongruence heorem (hm. 5.5) c orresponding parts of congruent triangles are congruent. 5 hapter 5 ongruent riangles Laurie s Notes eacher ctions iscuss the definition and vocabulary associated with an isosceles triangle. lways-ometimes-never rue: n equilateral triangle is an isosceles triangle. always true iscuss a plan for proving the ase ngles heorem. Give students time to try the proof on their own before discussing as a class. iscuss what the proof would look like if rigid motions were used. ommon Misconception: e sure students understand that there is more than one order in which the steps of a proof can be written. 5 hapter 5

36 Using the ase ngles heorem In,. Name two congruent angles. OLUION, so by the ase ngles heorem,. Monitoring rogress opy and complete the statement. 1. If HG HK, then.. If KHJ KJH, then. Help in nglish and panish at igideasmath.com G H K J nglish Language Learners Notebook evelopment Have students record the theorems and corollaries on pages 5 and 53 in their notebooks. or each statement, include a sketch and an example. xtra xample 1 In the figure,. Name two congruent sides. ecall that an equilateral triangle has three congruent sides. ING he corollaries state that a triangle is equilateral if and only if it is equiangular. orollaries orollary 5. orollary to the ase ngles heorem If a triangle is equilateral, then it is equiangular. roof x. 37, p. 58; x. 10, p. 353 orollary 5.3 orollary to the onverse of the ase ngles heorem If a triangle is equiangular, then it is equilateral. roof x. 39, p. 58 ind the measures of,, and. inding Measures in a riangle xtra xample ind the value of x. x = 30 x OLUION he diagram shows that is equilateral. o, by the orollary to the ase ngles heorem, is equiangular. o, m = m = m. 3(m ) = 180 riangle um heorem (heorem 5.1) m = 60 ivide each side by 3. MONIOING OG NW 1. HGK, HKG. KH, KJ 3. 5 U 5 he measures of,, and are all 60. Monitoring rogress 3. ind the length of for the triangle at the left. Help in nglish and panish at igideasmath.com ection 5.4 quilateral and Isosceles riangles 53 Laurie s Notes eacher ctions You could change xample 1. Have students draw an isosceles triangle. Mark the congruent angles. Have students name the congruent sides. If a conditional statement is true, then is its converse true? not always Write the two corollaries. xplain that they could be combined into one statement called a biconditional. esponse Logs: ight now I know ection

37 xtra xample 3 ind the values of x and y. 8 x = 15, y = 7 8 x 4y Using Isosceles and quilateral riangles onstructing an quilateral riangle onstruct an equilateral triangle that has side lengths congruent to. Use a compass and straightedge. OLUION tep 1 tep tep 3 tep 4 opy a segment opy. raw an arc raw an arc with center and radius. raw an arc raw an arc with center and radius. Label the intersection of the arcs from teps and 3 as. raw a triangle raw. ecause and are radii of the same circle,. ecause and are radii of the same circle,. y the ransitive roperty of ongruence (heorem.1),. o, is equilateral. ind the values of x and y in the diagram. Using Isosceles and quilateral riangles K y 4 L x + 1 OMMON O You cannot use N to refer to LNM because three angles have N as their vertex. N M OLUION tep 1 ind the value of y. ecause KLN is equiangular, it is also equilateral and KN KL. o, y = 4. tep ind the value of x. ecause LNM LMN, LN LM, and LMN is isosceles. You also know that LN = 4 because KLN is equilateral. LN = LM efinition of congruent segments 4 = x + 1 ubstitute 4 for LN and x + 1 for LM. 3 = x ubtract 1 from each side. 54 hapter 5 ongruent riangles Laurie s Notes eacher ctions Have students do the construction of the equilateral triangle. sk, How would you construct an isosceles triangle that is not equilateral? Listen for correct reasoning. If time permits, have students construct equilateral and isosceles triangles using construction software. urn and alk: ose the problem in xample 3. How can you solve for x and y in this diagram? Give partners time to work through the problem. Use opsicle ticks to solicit solutions. M3: Have students critique the reasoning used by their peers. 54 hapter 5

38 olving a Multi-tep roblem In the lifeguard tower, and. xtra xample 4 In the diagram, and. OMMON O When you redraw the triangles so that they do not overlap, be careful to copy all given information and labels correctly. 1 a. xplain how to prove that. b. xplain why is isosceles. OLUION 3 a. raw and label and so that they do not overlap. You can see that,, and. o, by the ongruence heorem (heorem 5.5),. 3 b. rom part (a), you know that 1 because corresponding parts of congruent triangles are congruent. y the onverse of the ase ngles heorem,, and is isosceles. Monitoring rogress 4. ind the values of x and y in the diagram. 5. In xample 4, show that. 4 1 ection 5.4 quilateral and Isosceles riangles 55 4 Help in nglish and panish at igideasmath.com y x a. xplain how to prove that. Given:. ecause, is an isosceles triangle. y the ase ngles heorem (hm. 5.6),. o, by heorem (hm. 5.5). b. xplain why is isosceles. rom part (a), because corresponding parts of s are. o is isosceles by definition. MONIOING OG NW 4. x = 60, y = rom xample 4, you know that and 1. It is stated that and. y the definition of congruent angles, m 1 = m and m = m. lso, by the ngle ddition ostulate (ost. 1.4), m 1 + m = m and m + m = m. y substituting m 1 for m and m for m, you get m 1 + m = m. hen, by the ransitive roperty of quality, m 1 + m = m 1 + m. o, by the ubtraction roperty of quality, m = m. ecause by the definition of congruent angles, you can conclude that by the ongruence heorem (hm. 5.5). Laurie s Notes eacher ctions eaching ip: When triangles overlap in a diagram, it is helpful to make a new diagram with the triangles not overlapping. eferencing angles by numbers versus three letters is easier to see and say. iscuss the plan for the solution, and then have students work independently to write a solution. losure esponse Logs: elect from When I am writing a proof or omorrow I need to find out or I wish I could. ection

39 ssignment Guide and Homework heck IGNMN asic: 1,, 3 19 odd,, 3, 38, 40, 4 44 verage: 1,, 4 16 even, 19 3, 34, 38, 40, 4 44 dvanced: 1,, 8 16 even, 19 30, 36 38, HOMWOK HK asic: 5, 9, 11, 13, 3 verage: 6, 8, 14, 3, 38 dvanced: 14,, 4, 37, 41 NW 1. he vertex angle is the angle formed by the congruent sides, or legs, of an isosceles triangle.. he base angles of an isosceles triangle are opposite the congruent sides, and they are congruent by the ase ngles heorem (hm. 5.6). 3., ; ase ngles heorem (hm. 5.6) 4., ; ase ngles heorem (hm. 5.6) 5., ; onverse of the ase ngles heorem (hm. 5.7) 6., ; onverse of the ase ngles heorem (hm. 5.7) 7. x = 1 8. x = x = x = x = 79, y = 1. ample answer: 5.4 xercises Vocabulary and ore oncept heck ynamic olutions available at igideasmath.com 1. VOULY escribe how to identify the vertex angle of an isosceles triangle.. WIING What is the relationship between the base angles of an isosceles triangle? xplain. Monitoring rogress and Modeling with Mathematics In xercises 3 6, copy and complete the statement. tate which theorem you used. (ee xample 1.) 3. If, then. 4. If, then. 5. If, then. 6. If, then. In xercises 7 10, find the value of x. (ee xample.) x 1 x x M L N x 11. MOLING WIH MHMI he dimensions of a sports pennant are given in the diagram. ind the values of x and y. 79 W x 5 y 1. MOLING WIH MHMI logo in an advertisement is an equilateral triangle with a side length of 7 centimeters. ketch the logo and give the measure of each side. In xercises 13 16, find the values of x and y. (ee xample 3.) y y x x x 5 y y 4 y 40 ONUION In xercises 17 and 18, construct an equilateral triangle whose sides are the given length inches inches 19. O NLYI escribe and correct the error in finding the length of. 5 6 x ecause,. o, = 6. 7 cm euse educe ecycle 7 cm 13. x = 60, y = x = 0, y = x = 30, y = x = 7, y = cm 56 hapter 5 ongruent riangles in. 19. When two angles of a triangle are congruent, the sides opposite the angles are congruent; ecause,. o, = 5. 3 in. 56 hapter 5

40 0. OLM OLVING he diagram represents part of the exterior of the ow ower in algary, lberta, anada. In the diagram, and are congruent equilateral triangles. (ee xample 4.) a. xplain why is isosceles. b. xplain why. c. how that and are congruent. d. ind the measure of. 1. INING N In the pattern shown, each small triangle is an equilateral triangle with an area of 1 square unit. a. xplain how you know that any triangle made out of equilateral triangles is equilateral. b. ind the areas of the first four triangles in the pattern. c. escribe any patterns in the areas. redict the area of the seventh triangle in the pattern. xplain your reasoning.. ONING he base of isosceles XYZ is YZ. What can you prove? elect all that apply. XY XZ X Y Y Z YZ ZX In xercises 3 and 4, find the perimeter of the triangle in. (4x + 1) in (x + 4) in. green blue riangle (1 x) in. rea 1 square unit (x 3) in. (x + 5) in. yellow orange MOLING WIH MHMI In xercises 5 8, use the diagram based on the color wheel. he 1 triangles in the diagram are isosceles triangles with congruent vertex angles. green blue bluepurple bluegreen yellowgreen redpurple yellow purple redorange orange yelloworange red 5. omplementary colors lie directly opposite each other on the color wheel. xplain how you know that the yellow triangle is congruent to the purple triangle. 6. he measure of the vertex angle of the yellow triangle is 30. ind the measures of the base angles. 7. race the color wheel. hen form a triangle whose vertices are the midpoints of the bases of the red, yellow, and blue triangles. (hese colors are the primary colors.) What type of triangle is this? 8. Other triangles can be formed on the color wheel that are congruent to the triangle in xercise 7. he colors on the vertices of these triangles are called triads. What are the possible triads? 9. IIL HINKING re isosceles triangles always acute triangles? xplain your reasoning. 30. IIL HINKING Is it possible for an equilateral triangle to have an angle measure other than 60? xplain your reasoning. 31. MHMIL ONNION he lengths of the sides of a triangle are 3t, 5t 1, and t + 0. ind the values of t that make the triangle isosceles. xplain your reasoning. 3. MHMIL ONNION he measure of an exterior angle of an isosceles triangle is x. Write expressions representing the possible angle measures of the triangle in terms of x. 33. WIING xplain why the measure of the vertex angle of an isosceles triangle must be an even number of degrees when the measures of all the angles of the triangle are whole numbers. ection 5.4 quilateral and Isosceles riangles 57 redpurple bluepurple purple equiangular equilateral yelloworange yellowgreen bluegreen redorange red 8. yellow-orange, red-purple, and blue-green; green, orange, and purple; red-orange, bluepurple, and yellow-green 9. no; he two sides that are congruent can form an obtuse angle or a right angle. 30. no; he sum of the angles of a triangle is always 180. o, if all three angles are congruent, then they will always be = ee dditional nswers. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 0. a. ecause and are congruent and equilateral, you know that. o, is isosceles. b. ecause is isosceles, by the ase ngles heorem (hm. 5.6). c. y the eflexive roperty of ongruence (hm..1),. ecause and are congruent and equilateral, and also equiangular by the orollary to the ase ngles heorem (or. 5.), you can conclude that. lso, as explained in part (a). o, by the ongruence heorem (hm. 5.5),. d a. ach edge is made out of the same number of sides of the original equilateral triangle. b. 1 square unit, 4 square units, 9 square units, 16 square units c. riangle 1 has an area of 1 = 1, riangle has an area of = 4, riangle 3 has an area of 3 = 9, and so on. o, by inductive reasoning, you can predict that riangle n has an area of n ; 49 square units; n = 7 = 49., in in. 5. y the eflexive roperty of ongruence (hm..1), the yellow triangle and the yelloworange triangle share a congruent side. ecause the triangles are all isosceles, by the ransitive roperty of ongruence (hm..1), the yellow-orange triangle and the orange triangle share a side that is congruent to the one shared by the yellow triangle and the yelloworange triangle. his reasoning can be continued around the wheel, so the legs of the isosceles triangles are all congruent. ecause you are given that the vertex angles are all congruent, you can conclude that the yellow triangle is congruent to the purple triangle by the ongruence heorem (hm. 5.5). ection

41 NW 34. a. XVY, UXV; WUX XVY because they are both vertex angles of congruent isosceles triangles. lso, m UXV + m VXY = m UXY by the ngle ddition ostulate (ost. 1.4), and m UXY = m WUX + m UWX by the xterior ngle heorem (hm. 5.). o, by the ransitive roperty of quality, m UXV + m VXY = m WUX + m UWX. lso, m UWX = m VXY because they are base angles of congruent isosceles triangles. y substituting m UWX for m VXY, you get m UXV + m UWX = m WUX + m UWX. y the ubtraction roperty of quality, m UXV = m WUX, so UXV WUX. b. 8 m ee dditional nswers. 34. OLM OLVING he triangular faces of the peaks on a roof are congruent isosceles triangles with vertex angles U and V. 6.5 m U W 8 m X Y a. Name two angles congruent to WUX. xplain your reasoning. b. ind the distance between points U and V. 35. OLM OLVING boat is traveling parallel to the shore along. When the boat is at point, the captain measures the angle to the lighthouse as 35. fter the boat has traveled.1 miles, the captain measures the angle to the lighthouse to be 70. V.1 mi a. ind L. xplain your reasoning. b. xplain how to find the distance between the boat and the shoreline. 36. HOUGH OVOKING he postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do all equiangular triangles have the same angle measures? Justify your answer. L 37. OVING OOLLY rove that the orollary to the ase ngles heorem (orollary 5.) follows from the ase ngles heorem (heorem 5.6). 38. HOW O YOU I? You are designing fabric purses to sell at the school fair. 100 a. xplain why. b. Name the isosceles triangles in the purse. c. Name three angles that are congruent to. 39. OVING OOLLY rove that the orollary to the onverse of the ase ngles heorem (orollary 5.3) follows from the onverse of the ase ngles heorem (heorem 5.7). 40. MKING N GUMN he coordinates of two points are (0, 6) and U(6, 0). Your friend claims that points, U, and V will always be the vertices of an isosceles triangle when V is any point on the line y = x. Is your friend correct? xplain your reasoning. 41. OO Use the diagram to prove that is equilateral. Given is equilateral. rove is equilateral. Mini-ssessment 1. ind the value of x. L Maintaining Mathematical roficiency eviewing what you learned in previous grades and lessons Use the given property to complete the statement. (ection.5) 4. eflexive roperty of ongruence (heorem.1): 43. ymmetric roperty of ongruence (heorem.1): If, then JK. 44. ransitive roperty of ongruence (heorem.1): If, and UV, then. M 76 (3x + 1) N 58 hapter 5 ongruent riangles x = 5 Use the figure. 10 5x. ind the length of ind m ind the value of x. 6 If students need help... esources by hapter ractice and ractice uzzle ime tudent Journal ractice ifferentiating the Lesson kills eview Handbook If students got it... esources by hapter nrichment and xtension umulative eview tart the next ection 58 hapter 5

42 What id You Learn? ore Vocabulary interior angles, p. 33 exterior angles, p. 33 corollary to a theorem, p. 35 corresponding parts, p. 40 ore oncepts lassifying riangles by ides, p. 3 lassifying riangles by ngles, p. 3 heorem 5.1 riangle um heorem, p. 33 heorem 5. xterior ngle heorem, p. 34 orollary 5.1 orollary to the riangle um heorem, p. 35 Identifying and Using orresponding arts, p. 40 heorem 5.3 roperties of riangle ongruence, p. 41 heorem 5.4 hird ngles heorem, p. 4 Mathematical ractices 1. In xercise 37 on page 37, what are you given? What relationships are present? What is your goal?. xplain the relationships present in xercise 3 on page 44. legs (of an isosceles triangle), p. 5 vertex angle (of an isosceles triangle), p. 5 base (of an isosceles triangle), p. 5 base angles (of an isosceles triangle), p escribe at least three different patterns created using triangles for the picture in xercise 0 on page 57. tudy kills Visual Learners heorem 5.5 ide-ngle-ide () ongruence heorem, p. 46 heorem 5.6 ase ngles heorem, p. 5 heorem 5.7 onverse of the ase ngles heorem, p. 5 orollary 5. orollary to the ase ngles heorem, p. 53 orollary 5.3 orollary to the onverse of the ase ngles heorem, p. 53 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. You are given a diagram of the triangle made from the segments that connect each person to the other two, along with the length of each segment; he people are all standing on the same stage (plane), so the points are coplanar; You are asked to classify the triangle by its sides and by measuring its angles.. here is a pair of congruent triangles, so all pairs of corresponding sides and angles are congruent. y the riangle um heorem (hm. 5.1), the three angles in LMN have measures that add up to 180. You are given two measures, so you can find the third using this theorem. he measure of is equal to the measure of its corresponding angle, L. he measure of is equal to the measure of its corresponding angle, N. Once you write this system of equations, you can solve for the values of the variables. 3. ample answer: a large triangle made up of 9 small triangles, a hexagon, a parallelogram raw a picture of a word problem. raw a picture of a word problem before starting to solve the problem. You do not have to be an artist. When making a review card for a word problem, include a picture. his will help you recall the information while taking a test. Make sure your notes are visually neat for easy recall. 59 hapter 5 59

43 NW corresponding sides:,, ; corresponding ngles:,, ; ample answer: 5. corresponding sides: WX, XY, YZ, WZ ; corresponding angles: W, X, Y, Z; ample answer: XYZW 6. no; he congruent angles are not the included angle ee dditional nswers. 9. WVX, WXV; ase ngles heorem (hm. 5.6) 10. XZY, XYZ; ase ngles heorem (hm. 5.6) 11. ZV, ZX ; onverse of the ase ngles heorem (hm. 5.7) 1. ZX, ZY ; onverse of the ase ngles heorem (hm. 5.7) 13. x = 13, y = x = 5, y = 0 15., a. right, obtuse, acute, equiangular b. equilateral, scalene, isosceles c. ee dditional nswers uiz ind the measure of the exterior angle. (ection 5.1) x. (5x + ) y Identify all pairs of congruent corresponding parts. hen write another congruence statement for the polygons. (ection 5.) WXYZ ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.5). If so, write a proof. If not, explain why. (ection 5.3) 6x 6., 7. GH, KHJ 8. LM, NM opy and complete the statement. tate which theorem you used. (ection 5.4) 9. If VW WX, then. 10. If XZ XY, then. 11. If ZVX ZXV, then. 1. If XYZ ZXY, then. ind the values of x and y. (ection 5. and ection 5.4) (5y 7) ft 9 38 ft G H K J 4 3. Y X 5x 1 6y (15x + 34) L V W W M Z Z 9 (1x + 6) N X Y (x + ) In a right triangle, the measure of one acute angle is 4 times the difference of the measure of the other acute angle and 5. ind the measure of each acute angle in the triangle. (ection 5.1) 16. he figure shows a stained glass window. (ection 5.1 and ection 5.3) a. lassify triangles 1 4 by their angles. b. lassify triangles 4 6 by their sides. c. Is there enough information given to prove that 7 8? If so, label the vertices and write a proof. If not, determine what additional information is needed. 60 hapter 5 ongruent riangles 60 hapter 5

44 Laurie s Notes Overview of ection 5.5 Introduction ide-ide-ide is usually the first method presented for proving two triangles congruent, and it is usually a postulate, with the remaining methods being presented and proven as theorems. he approach used in this chapter is to begin with the ide-ngle-ide () ongruence heorem (hm. 5.5). In this lesson, ide-ide-ide is presented as a theorem and proven using a transformational approach. third method, Hypotenuse-Leg (HL), is presented as a theorem in this lesson. ifferent approaches are used in the proof examples in the formal lesson: transformational, two-column, and paragraph. tudents should be comfortable with all three styles. new construction is presentedhow to copy a triangle using the ide-ide-ide () ongruence heorem. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. nother Way he classic, traditional approach to introducing the ongruence heorem is to use objects such as coffee stirrers to represent the sides of a triangle. Using stirrers of lengths 4, 5, and 6 inches, there is only one triangle that can be made. acing uggestion ake time for students to explore and discover the congruence relationship using dynamic geometry software, and then transition to the formal lesson. ection

45 ommon ore tate tandards HG-O..8 xplain how the criteria for triangle congruence ( ) follow from the definition of congruence in terms of rigid motions. HG-MG..1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). HG-MG..3 pply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Laurie s Notes xploration Motivate sk students to consider the following statements and whether the two triangles would be congruent. ll equilateral triangles are congruent. no ll equiangular triangles are congruent. no ll isosceles triangles with a vertex angle of 40 are congruent. no ll isosceles triangles with legs of 4 inches and a vertex angle of 40 are congruent. yes ll scalene triangles with side lengths 3 inches, 4 inches, and 5 inches are congruent. tudents may say yes to this one. Indeed it is yes and that will be proven in this lesson. xploration he ide-ide-ide () ongruence heorem (hm. 5.8) is the second of several theorems to show that less than six pieces of information are needed to prove that two triangles are congruent. his exploration may not be a familiar approach to verifying the triangle congruence method, so become familiar with the construction steps and potential for extensions. xploration 1 M5 Use ppropriate ools trategically: tudents should be familiar enough with the software to follow the construction described. he software will label points in the construction process that may differ from how the diagram and table are labeled. ell students to make adjustments as needed. he ordered pairs for points and will differ for each student, but the lengths of the three sides are still, 3, and 4 units, and the three missing angles will be the same measure. In constructing, students should try each orientation, clockwise and counterclockwise. re the results the same? Have students discuss the results of their explorations and make conjectures. xtension: epeat the construction for two circles of different radii or for a different length of. ommunicate Your nswer M3 onstruct Viable rguments and ritique the easoning of Others: tudents should conclude that when three sides of one triangle are congruent to the three sides of another triangle, the triangles are congruent. onnecting to Next tep he exploration allows students to discover and make sense of the congruence relationship that will be presented formally in the lesson. -61 hapter 5

46 UING OOL GILLY 5.5 o be proficient in math, you need to use technology to help visualize the results of varying assumptions, explore consequences, and compare predictions with data. roving riangle ongruence by ssential uestion What can you conclude about two triangles when you know the corresponding sides are congruent? rawing riangles Work with a partner. Use dynamic geometry software. a. onstruct circles with radii of units and 3 units centered at the origin. Label the origin. hen draw of length 4 units. b. Move so that is on the smaller circle and is on the larger circle. hen draw. c. xplain why the side lengths of are, 3, and 4 units. d. ind m, m, and m. e. epeat parts (b) and (d) several times, moving to different locations. Keep track of your results by copying and completing the table below. What can you conclude? m m m 1. (0, 0) 3 4. (0, 0) (0, 0) (0, 0) (0, 0) ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work b. heck students work. c. = because has one endpoint at the origin and one endpoint on a circle with a radius of units. = 3 because has one endpoint at the origin and one endpoint on a circle with a radius of 3 units. = 4 because it was created that way. d. m = , m = 46.61, m = 8.96 e. heck students work; If two triangles have three pairs of congruent sides, then they will have three pairs of congruent angles.. he corresponding angles are also congruent and therefore, the triangles are congruent. 3. Use rigid transformations to map triangles. ommunicate Your nswer. What can you conclude about two triangles when you know the corresponding sides are congruent? 3. How would you prove your conclusion in xploration 1(e)? ection 5.5 roving riangle ongruence by 61 ection

47 ifferentiated Instruction Inclusion emind students that the sides of a triangle are segments. egments can be congruent when their lengths are equal. Likewise, the angles in a triangle are congruent when their measures are equal. oint out that the and ongruence heorems require that pairs of congruent corresponding parts be identified. his will help students when proving the congruence of two triangles. 5.5 Lesson ore Vocabulary legs, p. 64 hypotenuse, p. 64 revious congruent figures rigid motion What You Will Learn Use the ide-ide-ide () ongruence heorem. Use the Hypotenuse-Leg (HL) ongruence heorem. Using the ide-ide-ide ongruence heorem heorem heorem 5.8 ide-ide-ide () ongruence heorem If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If,, and, then. ide-ide-ide () ongruence heorem Given,, rove irst, translate so that point maps to point, as shown below. his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below ecause, the rotation maps point to point. o, this rotation maps to. raw an auxiliary line through points and. his line creates 1,, 3, and 4, as shown at the left. ecause, is an isosceles triangle. ecause, is an isosceles triangle. y the ase ngles heorem (hm. 5.6), 1 3 and 4. y the definition of congruence, m 1 = m 3 and m = m 4. y construction, m = m 1 + m and m = m 3 + m 4. You can now use the ubstitution roperty of quality to show m = m. m = m 1 + m ngle ddition ostulate (ostulate 1.4) = m 3 + m 4 ubstitute m 3 for m 1 and m 4 for m. = m ngle ddition ostulate (ostulate 1.4) y the definition of congruence,. o, two pairs of sides and their included angles are congruent. y the ongruence heorem (hm. 5.5),. o, a composition of rigid motions maps to. ecause a composition of rigid motions maps to and a composition of rigid motions maps to, a composition of rigid motions maps to. o,. 6 hapter 5 ongruent riangles Laurie s Notes eacher ctions he ide-ide-ide () ongruence heorem should make sense following the exploration. iscuss the first few steps in the proof. OMMON O Once has been translated and rotated, students want to reflect onto. ecause no relationship is known about the angles yet, would map onto only if was a valid congruence method, but that is what we are proving and that would be circular reasoning. Work through the remaining steps of the proof as shown. 6 hapter 5

48 Using the ongruence heorem nglish Language Learners L Write a proof. NL, KM NM Given KL K air ctivity N rove KLM NLM M OLUION MN ON NL 1. KL 1. Given. Given. KM NM 3. LM LM 3. eflexive roperty of ongruence (hm..1) 4. KLM NLM 4. ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com Have one student draw a triangle and record the approximate lengths of its sides. hen, have the student give the lengths of the sides to his or her partner. Have the partner draw a triangle having the same side lengths, and then compare the two triangles. sk students to make a conjecture about the relationship between the triangles that have three pairs of congruent sides. ecide whether the congruence statement is true. xplain your reasoning. 1. G HJK. J 3 G H 7 4 xtra xample 1 Write a proof., Given rove 9 7 K 3. olving a eal-life roblem xplain why the bench with the diagonal support is stable, while the one without the support can collapse. tatements (easons) OLUION he bench with the diagonal support forms triangles with fixed side lengths. y the ongruence heorem, these triangles cannot change shape, so the bench is stable. he bench without the diagonal support is not stable because there are many possible quadrilaterals with the given side lengths. Monitoring rogress Help in nglish and panish at igideasmath.com, (Given) 1.. (eflex. rop. hm..1) 3. ( hm. 5.8) xtra xample etermine whether the hexagon is stable. xplain your reasoning. etermine whether the figure is stable. xplain your reasoning ection 5.5 hs_geo_pe_0505.indd 63 Laurie s Notes eacher ctions roving riangle ongruence by 63 1/19/15 10:30 M urn and alk: ose xample 1. What additional information is needed to prove the triangles congruent? hink-air-hare: Have students answer uestions 1 3, and then share and discuss as a class. sk students whether any of them have seen the cross brace that builders use when a wall has only the vertical two-by-four studs. he brace prevents the wall from leaning left or right. he application in xample is similar. he diagonal supports form two triangles and a quadrilateral. he figure is not stable because there are many possible quadrilaterals with the given side lengths. MONIOING OG NW 1. yes; rom the diagram markings, HJ, G JK, and G HK. o, G HJK by the ongruence heorem (hm. 5.8). 6. ee dditional nswers. ection 5.5 hscc_geo_te_0505.indd /1/15 3:03 M

49 opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OLUION tep 1 tep tep 3 tep 4 onstruct a side onstruct so that it is congruent to. raw an arc Open your compass to the length. Use this length to draw an arc with center. raw an arc raw an arc with radius and center that intersects the arc from tep. Label the intersection point. raw a triangle raw. y the ongruence heorem,. Using the Hypotenuse-Leg ongruence heorem You know that and are valid methods for proving that triangles are congruent. What about? In general, is not a valid method for proving that triangles are congruent. In the triangles below, two pairs of sides and a pair of angles not included between them are congruent, but the triangles are not congruent. leg hypotenuse leg While is not valid in general, there is a special case for right triangles. In a right triangle, the sides adjacent to the right angle are called the legs. he side opposite the right angle is called the hypotenuse of the right triangle. heorem heorem 5.9 Hypotenuse-Leg (HL) ongruence heorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. If,, and m = m = 90, then. roof x. 38, p. 470; igideasmath.com 64 hapter 5 ongruent riangles Laurie s Notes hs_geo_pe_0505.indd 64 eacher ctions How could you construct a copy of a triangle using the ongruence heorem (hm. 5.8)? Listen for steps shown in the construction. robing uestion: You know that and are two methods for proving triangles congruent. o you think is a valid method? nswers will vary. sk students to draw measuring 40. onstruct side with length 4 inches. Now construct side with length 3 inches. wo possible triangles can be formed, meaning is not a valid triangle congruence method, except for right triangles. iscuss the Hypotenuse-Leg (HL) ongruence heorem. 1/19/15 10:30 M 64 hapter 5

50 UY I If you have trouble matching vertices to letters when you separate the overlapping triangles, leave the triangles in their original orientations. W Z Y Z X Y Write a proof. Given WY XZ, WZ ZY, XY Z Y rove WYZ XZY Using the Hypotenuse-Leg ongruence heorem OLUION edraw the triangles so they are side by side with corresponding parts in the same position. Mark the given information in the diagram. MN H 1. WY XZ. WZ ZY, XY ZY ON 1. Given. Given 3. Z and Y are right angles. 3. efinition of lines 4. WYZ and XZY are right triangles. 4. efinition of a right triangle L 5. ZY YZ 5. eflexive roperty of ongruence (hm..1) 6. WYZ XZY 6. HL ongruence heorem Using the Hypotenuse-Leg ongruence heorem he television antenna is perpendicular to the plane containing points,,, and. ach of the cables running from the top of the antenna to,, and has the same length. rove that,, and are congruent. Given,,, rove OLUION You are given that and. o, and are right angles by the definition of perpendicular lines. y definition, and are right triangles. You are given that the hypotenuses of these two triangles, and, are congruent. lso, is a leg for both triangles, and by the eflexive roperty of ongruence (hm..1). o, by the Hypotenuse-Leg ongruence heorem,. You can use similar reasoning to prove that. W Z W Z Y X Y X Y Z xtra xample 3 Write a proof. Given, and are right angles. rove tatements (easons) 1. (Given). and are right angles. (Given) 3. and are right triangles. (ef. of rt. ) 4. (eflex. rop. of hm..1) 5. (HL hm. 5.9) xtra xample 4 wo windows in a building are right triangles, with WX XY. rove that the triangles are congruent. Given WXZ and YXZ are right triangles, WX XY. rove WXZ YXZ W X Z Y Laurie s Notes o, by the ransitive roperty of riangle ongruence (hm. 5.3),. Monitoring rogress Use the diagram. 7. edraw and side by side with corresponding parts in the same position. 8. Use the information in the diagram to prove that. eacher ctions Help in nglish and panish at igideasmath.com ection 5.5 roving riangle ongruence by 65 M4 Model with Mathematics: o complete the proof in xample 3, a useful technique is to redraw the diagram so that the triangles do not overlap. M3: Note the sequence of steps in this proof. he perpendicular segments give you right angles and then you have right triangles. or xample 4, you might have students take the paragraph proof and rewrite it as a two-column proof. losure esponse Logs: elect from What confused me the most was or I will understand this better if I or I am feeling good about. You are given that WX XY and that the triangles are right triangles. o, the hypotenuses of the triangles are congruent. y the eflexive roperty of (hm..1), XZ XZ. o, by the HL heorem (hm. 5.9), WXZ YXZ. MONIOING OG NW ee dditional nswers. ection

51 ssignment Guide and Homework heck IGNMN asic: 1,, 3 3 odd, 30, 31, verage: 1, 0 even, 1,, 4, 30, 31, 34, dvanced: 1,, 10 0 even, 1, 8 40 HOMWOK HK asic: 3, 5, 7, 13, 31 verage: 8, 10, 14, 16, 36 dvanced: 10, 14, 16, 8, 36 NW 1. hypotenuse. 5.5 xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. OML H NN he side opposite the right angle is called the of the right triangle.. WHIH ON ON LONG? Which triangle s legs do not belong with the other three? xplain your reasoning. Monitoring rogress and Modeling with Mathematics In xercises 3 and 4, decide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (heorem 5.8). xplain. 3., 4., 9. G 10. JKL LJM G K J L M his is the only triangle with legs that are not the legs of a right triangle. 3. yes;,, 4. no; You cannot tell for sure from the diagram whether or not and are congruent. 5. yes; and are right angles,, 6. no; he hypotenuses are not marked as congruent. 7. no; You are given that,, and. o, it should say by the ongruence heorem (hm. 5.8). 8. yes; You are given that and. lso, by the eflexive roperty of ongruence (hm..1). o, by the ongruence heorem (hm. 5.8). 9. yes; You are given that G and G. lso, by the eflexive roperty of ongruence (hm..1). o, G by the ongruence heorem (hm. 5.8). 10. no; You are given that JK KL LM MJ. lso, JL JL by the eflexive roperty of ongruence (hm..1). o, it should say JKL LMJ or JKL JML by the ongruence heorem (hm. 5.8). In xercises 5 and 6, decide whether enough information is given to prove that the triangles are congruent using the HL ongruence heorem (heorem 5.9). xplain. 5., 6., In xercises 7 10, decide whether the congruence statement is true. xplain your reasoning. (ee xample 1.) hapter 5 ongruent riangles hs_geo_pe_0505.indd yes; he diagonal supports in this figure form triangles with fixed side lengths. y the ongruence heorem (hm. 5.8), these triangles cannot change shape, so the figure is stable. 1. no; he support in this figure forms two quadrilaterals, which are not stable because there are many possible quadrilaterals with the given side lengths ee dditional nswers. In xercises 11 and 1, determine whether the figure is stable. xplain your reasoning. (ee xample.) In xercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. hen write a proof. (ee xample 3.) 13. Given,, rove 14. Given G is the midpoint of H, G GI, and H are right angles. rove G HIG I G H 1/19/15 10:31 M 66 hapter 5

52 In xercises 15 and 16, write a proof. 15. Given LM JK, MJ KL rove LMJ JKL K L. MOLING WIH MHMI he distances between consecutive bases on a softball field are the same. he distance from home plate to second base is the same as the distance from first base to third base. he angles created at each base are 90. rove H H. (ee xample 4.) ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library J 16. Given WX VZ, WY VY, YZ YX rove VWX WVZ W M X third base () second base () first base () ynamic lassroom with ynamic Investigations NW ee dditional nswers. 17. V ONUION In xercises 17 and 18, construct a triangle that is congruent to using the ongruence heorem (heorem 5.8). Y Z home plate (H) 3. ONING o support a tree, you attach wires from the trunk of the tree to stakes in the ground, as shown in the diagram O NLYI escribe and correct the error in identifying congruent triangles. V U X Z UV XYZ by the ongruence heorem. 0. O NLYI escribe and correct the error in determining the value of x that makes the triangles congruent. x + 1 J 6x K M 4x + 4 L 3x 1 Y 6x = x + 1 4x = 1 x = MKING N GUMN Your friend claims that in order to use the ongruence heorem (heorem 5.8) to prove that two triangles are congruent, both triangles must be equilateral triangles. Is your friend correct? xplain your reasoning. J ection 5.5 roving riangle ongruence by 67 K a. What additional information do you need to use the HL ongruence heorem (heorem 5.9) to prove that JKL MKL? b. uppose K is the midpoint of JM. Name a theorem you could use to prove that JKL MKL. xplain your reasoning. 4. ONING Use the photo of the Navajo rug, where and. L M a. What additional information do you need to use the ongruence heorem (heorem 5.8) to prove that? b. What additional information do you need to use the HL ongruence heorem (heorem 5.9) to prove that? he order of the points in the congruence statement should reflect the corresponding sides and angles; UV ZYX by the ongruence heorem (hm. 5.8). 0. When you substitute 1 for x, 4 KL LM ; 6x = 4x + 4 and x + 1 = 3x 1; x =. 1. no; he sides of a triangle do not have to be congruent to each other, but each side of one triangle must be congruent to the corresponding side of the other triangle.. ee dditional nswers. 3. a. You need to know that the hypotenuses are congruent: JL ML. b. ongruence heorem (hm. 5.5); y definition of midpoint, JK MK. lso, LK LK, by the eflexive roperty of ongruence (hm..1), and JKL MKL by the ight ngles ongruence heorem (hm..3). 4. a. b. and are right angles. ection

53 NW 5. congruent 6. not congruent 7. congruent 8. not congruent 9. yes; Use the string to compare the lengths of the corresponding sides of the two triangles to determine whether ongruence heorem (hm. 5.8) applies ee dditional nswers. In xercises 5 8, use the given coordinates to determine whether. 5. (, ), (4, ), (4, 6), (5, 7), (5, 1), (13, 1) 6. (, 1), (3, 3), (7, 5), (3, 6), (8, ), (10, 11) 7. (0, 0), (6, 5), (9, 0), (0, 1), (6, 6), (9, 1) 8. ( 5, 7), ( 5, ), (0, ), (0, 6), (0, 1), (4, 1) 9. IIL HINKING You notice two triangles in the tile floor of a hotel lobby. You want to determine whether the triangles are congruent, but you only have a piece of string. an you determine whether the triangles are congruent? xplain. 30. HOW O YOU I? here are several theorems you can use to show that the triangles in the square pattern are congruent. Name two of them. 3. HOUGH OVOKING he postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are points on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, do you think that two triangles are congruent if their corresponding sides are congruent? Justify your answer. UING OOL In xercises 33 and 34, use the given information to sketch LMN and U. Mark the triangles with the given information. 33. LM MN, U, LM NM U 34. LM MN, U, LM, LN U 35. IIL HINKING he diagram shows the light created by two spotlights. oth spotlights are the same distance from the stage. G Mini-ssessment ecide whether the statement is true. xplain your reasoning MKING N GUMN Your cousin says that JKL is congruent to LMJ by the ongruence heorem (hm. 5.8). Your friend says that JKL is congruent to LMJ by the HL ongruence heorem (hm. 5.9). Who is correct? xplain your reasoning. J M K L a. how that. tate which theorem or postulate you used and explain your reasoning. b. re all four right triangles shown in the diagram congruent? xplain your reasoning. 36. MHMIL ONNION ind all values of x that make the triangles congruent. xplain. 5x 5x 4x + 3 3x + 10 Yes, HL heorem (hm. 5.9). WXY YZW X Maintaining Mathematical roficiency Use the congruent triangles. (ection 5.) 37. Name the segment in that is congruent to. 38. Name the segment in that is congruent to. 39. Name the angle in that is congruent to. eviewing what you learned in previous grades and lessons 40. Name the angle in that is congruent to. W Y 68 hapter 5 ongruent riangles Yes, heorem (hm. 5.8) 3. Z No, the HL heorem (hm. 5.9) applies only to right triangles, and the measures of and are not known. If students need help... esources by hapter ractice and ractice uzzle ime tudent Journal ractice ifferentiating the Lesson kills eview Handbook If students got it... esources by hapter nrichment and xtension umulative eview tart the next ection 68 hapter 5

54 Laurie s Notes Overview of ection 5.6 Introduction wo additional triangle congruence theorems are introduced in this lesson, ngle-ide-ngle () and ngle-ngle-ide (). Like the previous lessons, there is a blend of proof styles: transformations, two-column, and paragraph. he construction presented in this lesson is copying a triangle using the ongruence heorem. dditional examples with proving triangles congruent are presented, requiring students to recall previously proven theorems. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. nother Way low-tech version of investigating the validity of different congruence theorems is to draw triangle parts on pieces of transparency. Make several copies of each of the parts so that two [possibly] congruent triangles can be made. he angles should be drawn with long enough rays so that points of intersection can be found. My collection of angle pieces includes angles of 40, 60, and 80. My collection of side pieces includes lengths of centimeters, 3 centimeters, 4 centimeters, and 5 centimeters. YI: his collection of pieces is also used to show that sides of lengths, 3, and 5 do not form a triangle. acing uggestion Once students have worked the explorations, continue with the formal lesson. ection

55 ommon ore tate tandards HG-O..8 xplain how the criteria for triangle congruence (, ) follow from the definition of congruence in terms of rigid motions. Laurie s Notes xploration Motivate If possible, bring a wall mirror to class. tand in front of the mirror and describe the amount of your reflection that you see in the mirror. If I move closer to or farther away from the mirror, will the amount of reflection I see change? nswers will vary. xplain that in this lesson additional congruence theorems will be introduced, and in the exercises students will explore this question. xploration 1 his exploration helps students understand why the ide-ide-ngle () method is not a valid triangle-congruence theorem. note about this method was made on page 64, prior to introducing the HL ongruence heorem (hm. 5.9). M5 Use ppropriate ools trategically: tudents should be familiar enough with the software to follow the construction described. he software will label points in the construction process that may differ from how the diagram and table are labeled. ell students to make adjustments as needed. What do you know about the sides and angles of and? is in both triangles, and in is congruent to in ; is in both triangles. Is? xplain. no; he third sides of the two triangles are not congruent:. xploration tudents have already concluded that (hm. 5.5) and (hm. 5.8) are valid trianglecongruence theorems and that is not. lternate Method: You might have other approaches such as compass and protractor that students could use to explore ngle-ngle-ide (), ngle-ide-ngle (), and ngle- ngle-ngle () methods. ommunicate Your nswer M3 onstruct Viable rguments and ritique the easoning of Others: tudents should conclude that having three pieces of information about two triangles is not always sufficient to conclude that the triangles are congruent. onnecting to Next tep he exploration allows students to discover and make sense of why is not a valid method and to see that there may be other methods besides and to investigate. hese will be presented formally in the lesson. -69 hapter 5

56 5.6 ONUING VIL GUMN o be proficient in math, you need to recognize and use counterexamples. roving riangle ongruence by and ssential uestion What information is sufficient to determine whether two triangles are congruent? etermining Whether Is ufficient Work with a partner. a. Use dynamic geometry software to construct. onstruct the triangle so that vertex is at the origin, has a length of 3 units, and has a length of units. b. onstruct a circle with a radius of units centered at the origin. Locate point where the circle intersects. raw ection 5.6 roving riangle ongruence by and 69 3 ample oints (0, 3) (0, 0) (, 0) (0.77, 1.85) egments = 3 = 3.61 = = 1.38 ngle m = c. and have two congruent sides and a nonincluded congruent angle. Name them. d. Is? xplain your reasoning. e. Is sufficient to determine whether two triangles are congruent? xplain your reasoning. etermining Valid ongruence heorems Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. or those that are not valid, write a counterexample. xplain your reasoning. ossible ongruence heorem ommunicate Your nswer Valid or not valid? 3. What information is sufficient to determine whether two triangles are congruent? 4. Is it possible to show that two triangles are congruent using more than one congruence theorem? If so, give an example. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work. b. heck students work. c.,, and d. no; he third pair of sides are not congruent. e. no; hese two triangles provide a counterexample for. hey have two pairs of congruent sides and a pair of nonincluded congruent angles, but the triangles are not congruent.. ee dditional nswers. 3. In order to determine that two triangles are congruent, one of the following must be true. ll three pairs of corresponding sides are congruent (). wo pairs of corresponding sides and the pair of included angles are congruent (). wo pairs of corresponding angles and the pair of included sides are congruent (). wo pairs of corresponding angles and one pair of nonincluded sides are congruent (). he hypotenuses and one pair of corresponding legs of two right triangles are congruent (HL). 4. yes; ample answer: In the diagram, by the HL ongruence heorem (hm. 5.9), the ongruence heorem (hm. 5.8), and the ongruence heorem (hm. 5.5). ection

57 ifferentiated Instruction Kinesthetic tudents may wonder why there is not an riangle ongruence heorem. sk each student to use a protractor and straightedge to draw a triangle. hen have students compare their triangles. mphasize that three pairs of congruent corresponding angles is necessary but not sufficient to prove two triangles congruent. 5.6 Lesson ore Vocabulary revious congruent figures rigid motion What You Will Learn ngle-ide-ngle () ongruence heorem Given,, rove Use the and ongruence heorems. Using the and ongruence heorems heorem heorem 5.10 ngle-ide-ngle () ongruence heorem If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 70 irst, translate so that point maps to point, as shown below. 70 hapter 5 ongruent riangles his translation maps to. Next, rotate counterclockwise through so that the image of coincides with, as shown below. ecause, the rotation maps point to point. o, this rotation maps to. Now, reflect in the line through points and, as shown below. ecause points and lie on, this reflection maps them onto themselves. ecause a reflection preserves angle measure and, the reflection maps to. imilarly, because, the reflection maps to. he image of lies on and. ecause and only have point in common, the image of must be. o, this reflection maps to. ecause you can map to using a composition of rigid motions,. Laurie s Notes eacher ctions raw and and mark two pairs of angles congruent. What pair of sides would need to be congruent if ngle-ide-ngle () is a valid triangle-congruence method? the side between the angles urn and alk: How could you prove works using transformations? Give partners time to discuss the steps needed. e sure that students rotate to, mapping the congruent sides onto one another. 70 hapter 5

58 heorem heorem 5.11 ngle-ngle-ide () ongruence heorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. If,, and, then. roof p. 71 Given rove,, ngle-ngle-ide () ongruence heorem nglish Language Learners Words and bbreviations sk students to explain the importance of the order of the letters in the,, and ongruence heorems. xtra xample 1 an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. a. G and HG You are given and. y the hird ngles heorem (heorem 5.4),. You are given. o, two pairs of angles and their included sides are congruent. y the ongruence heorem,. G Identifying ongruent riangles an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. a. b. c. H Yes, heorem (hm. 5.11) b. M and M OMMON O You need at least one pair of congruent corresponding sides to prove two triangles are congruent. Laurie s Notes OLUION a. he vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. he triangles are congruent by the ongruence heorem. b. here is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. wo pairs of angles and their included sides are congruent. he triangles are congruent by the ongruence heorem. Monitoring rogress 1. an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. Help in nglish and panish at igideasmath.com ection 5.6 roving riangle ongruence by and 71 eacher ctions raw and and mark two pairs of angles congruent. What pair of sides would need to be congruent if ngle-ngle-ide () is a valid triangle-congruence method? a side not between the angles iscuss the outline of the proof for the ongruence heorem. hink-air-hare: Have students try the three parts in xample 1 independently before discussing as a class. esponse Logs: ight now I need to W X Z Y M No c. LM and NM M L N Yes, heorem (hm. 5.10) MONIOING OG NW 1. yes; ongruence heorem (hm. 5.10) ection

59 xtra xample Write a proof. Given H G, G rove H G opying a riangle Using onstruct a triangle that is congruent to using the ongruence heorem. Use a compass and straightedge. OLUION tep 1 tep tep 3 tep 4 H ample answer: tatements (easons) 1. H G, G (Given). H G (Vert. s hm..6) 3. G (lt. Int. s hm. 3.) 4. H G ( hm. 5.10) G MONIOING OG NW. ee dditional nswers. onstruct a side onstruct so that it is congruent to. onstruct an angle onstruct with vertex and side so that it is congruent to. Write a proof. Given, rove OLUION onstruct an angle onstruct with vertex and side so that it is congruent to. Using the ongruence heorem Label a point Label the intersection of the sides of and that you constructed in teps and 3 as. y the ongruence heorem,. MN ON 1. Given. lternate Interior ngles heorem (hm. 3.) 3. Given 4. Vertical ngles ongruence heorem (hm.6) ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com. In the diagram,,, and. rove. 7 hapter 5 ongruent riangles Laurie s Notes hs_geo_pe_0506.indd 7 eacher ctions onstructing angles at each endpoint of the segment can be challenging for students, particularly when is short. ncourage students to begin with a reasonably sized triangle to copy using the ongruence heorem. Whiteboarding: Have students work with partners to write a proof for xample. irculate to observe their approaches. M3: elect several students to share their proofs, demonstrating that the order of the steps may vary and that there are two possible methods ( and ). 1/19/15 10:3 M 7 hapter 5

60 Using the ongruence heorem Write a proof. Given H GK, and K are right angles. rove HG GKH OLUION MN 1. H GK ON 1. Given. GH HGK. lternate Interior ngles heorem (heorem 3.) 3. and K are right angles. 3. Given 4. K 4. ight ngles ongruence heorem (heorem.3) 5. HG GH 5. eflexive roperty of ongruence (heorem.1) 6. HG GKH 6. ongruence heorem Monitoring rogress Help in nglish and panish at igideasmath.com 3. In the diagram, U and VU. rove VU. H G K xtra xample 3 Write a proof. Use the diagram and the given information in xample. rove using the ongruence heorem. ample answer: tatements (easons) 1. (Given). (lt. Int. s hm. 3.) 3. (lt. Int. s hm. 3.) 4. (Given) 5. ( hm. 5.11) MONIOING OG NW 3. ee dditional nswers. U V oncept ummary riangle ongruence heorems You have learned five methods for proving that triangles are congruent. HL (right s only) wo sides and the included angle are congruent. ll three sides are congruent. he hypotenuse and one of the legs are congruent. wo angles and the included side are congruent. wo angles and a non-included side are congruent. In the xercises, you will prove three additional theorems about the congruence of right triangles: Hypotenuse-ngle, Leg-Leg, and ngle-leg. ection 5.6 roving riangle ongruence by and 73 Laurie s Notes eacher ctions OMMON O In xample 3, students may try to state GH KHG. hese are alternate interior angles for two sides not known to be parallel, G and HK. eaching ip: Use a color highlighter to extend the parallel sides H and GK. ransversal HG forms congruent alternate interior angles GH and HGK. losure esponse Logs: elect from: I feel like I made progress today or I m still not sure about or sticky part for me is. ection

61 ssignment Guide and Homework heck IGNMN asic: 1,, 3 1 odd, 8, 30, verage: 1, 4 even, 8 30, dvanced: 1,, 6 10 even, 14 8 even, 9, 30, 3 38 HOMWOK HK asic: 5, 7, 11, 15, 1 verage: 6, 8, 16, 18, 9 dvanced: 10, 16, 18, 9, 33 NW 1. oth theorems are used to prove that two triangles are congruent, and both require two pairs of corresponding angles to be congruent. In order to use the ongruence heorem (hm. 5.11), one pair of corresponding nonincluded sides must also be congruent. In order to use the ongruence heorem (hm. 5.10), the pair of corresponding included sides must be congruent.. You need to know that one pair of corresponding sides are congruent. 3. yes; ongruence heorem (hm. 5.11) 4. yes; ongruence heorem (hm 5.11) 5. no 6. yes; ongruence heorem (hm. 5.10) 7. ; L 8. ; L 9. yes; by the ongruence heorem (hm. 5.10) 10. no; he congruence statements follow the pattern, which is not sufficient to conclude that the triangles are congruent. 11. no; and do not correspond. 1. yes; by the ongruence heorem (hm. 5.11) xercises ynamic olutions available at igideasmath.com Vocabulary and ore oncept heck 1. WIING How are the ongruence heorem (heorem 5.11) and the ongruence heorem (heorem 5.10) similar? How are they different?. WIING You know that a pair of triangles has two pairs of congruent corresponding angles. What other information do you need to show that the triangles are congruent? Monitoring rogress and Modeling with Mathematics In xercises 3 6, decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. (ee xample 1.) 3., 4., 5. XYZ, JKL 6. V, UV X Y Z L K In xercises 7 and 8, state the third congruence statement that is needed to prove that GH LMN using the given theorem. 74 hapter 5 ongruent riangles G 7. Given GH MN, G M, H J L Use the ongruence heorem (hm. 5.11). 8. Given G LM, G M, Use the ongruence heorem (hm. 5.10). 15. In the congruence statement, the vertices should be in corresponding order; JKL GH by the ongruence heorem (hm. 5.10). 16. he included sides are congruent; VUX by the ongruence heorem (hm. 5.10). U M V N In xercises 9 1, decide whether you can use the given information to prove that. xplain your reasoning. 9.,, 10.,, 11.,, 1.,, ONUION In xercises 13 and 14, construct a triangle that is congruent to the given triangle using the ongruence heorem (heorem 5.10). Use a compass and straightedge O NLYI In xercises 15 and 16, describe and correct the error. 15. K H JKL HG L by the G ongruence J heorem. 16. X W V L J K VWX by the ongruence heorem. 74 hapter 5

62 OO In xercises 17 and 18, prove that the triangles are congruent using the ongruence heorem (heorem 5.10). (ee xample.). M is the midpoint of NL L NL N, NL M, M 17. Given 3. ngle-leg (L) ongruence heorem If an angle and a leg of a right triangle are congruent to an angle and a leg of a second right triangle, then the triangles are congruent. 4. ONING What additional information do KM KJ KH NH N N LKJ LNM NW H M J L M J 18. Given J K, JK KJ, rove 5. MHMIL ONNION his toy K J contains and. an you conclude that from the given angle measures? xplain. ynamic lassroom with ynamic Investigations M K L J K OO In xercises 19 and 0, prove that the triangles are congruent using the ongruence heorem (heorem 5.11). (ee xample 3.) m = (8x 3) 19. Given m = (4y 4) rove UW, X Z VW m = (5x + 10) XWV ZWU Z Y V m = (x 8) U m = (y 6) W 0. Given rove m = (3y + ) X 6. ONING Which of the following congruence statements are true? elect all that apply. NKM LMK, L N U UV NMK LKM L V XVW N W X V VWU K V VUW M OO In xercises 1 3, write a paragraph proof for the theorem about right triangles. 8. MKING N GUMN Your friend claims to be able to rewrite any proof that uses the ongruence heorem (hm. 5.11) as a proof that uses the ongruence heorem (hm. 5.10). Is this possible? xplain your reasoning. a right triangle are congruent to the legs of a second right triangle, then the triangles are congruent. hs_geo_pe_0506.indd 75 V ase ngles heorem (heorem 5.7). (Hint: raw an auxiliary line inside the triangle.). Leg-Leg (LL) ongruence heorem If the legs of ection 5.6 U 7. OVING HOM rove the onverse of the 1. Hypotenuse-ngle (H) ongruence heorem If an angle and the hypotenuse of a right triangle are congruent to an angle and the hypotenuse of a second right triangle, then the triangles are congruent. roving riangle ongruence by and 8. yes; When two triangles are congruent by the ongruence heorem (hm. 5.11), their third angles are congruent by the hird ngles heorem (hm. 5.4), so the ongruence heorem (hm. 5.10) will also apply. ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library you need to prove JKL MNL by the ongruence heorem (heorem 5.10)? NM ML rove ynamic eaching ools 75 1/19/15 10:3 M ee dditional nswers. 1. You are given two right triangles, so the triangles have congruent right angles by the ight ngles ongruence heorem (hm..3). ecause another pair of angles and a pair of corresponding nonincluded sides (the hypotenuses) are congruent, the triangles are congruent by the ongruence heorem (hm. 5.11).. You are given two right triangles, so the triangles have congruent right angles by the ight ngles ongruence heorem (hm..3). ecause both pairs of legs are congruent, and the congruent right angles are the included angles, the triangles are congruent by the ongruence heorem (hm. 5.5) 3. You are given two right triangles, so the triangles have congruent right angles by the ight ngles ongruence heorem (hm..3). here is also another pair of congruent corresponding angles and a pair of congruent corresponding sides. If the pair of congruent sides is the included side, then the triangles are congruent by the ongruence heorem (hm. 5.10). If the pair of congruent sides is a nonincluded pair, then the triangles are congruent by the ongruence heorem (hm. 5.11) yes; When x = 14 and y = 6, m = m = m = m = 80 and m = m = 0. his satisfies the riangle um heorem (hm. 5.1) for by both triangles. ecause the eflexive roperty of ongruence (hm..1), you can conclude that by the ongruence heorem (hm. 5.10) or the ongruence heorem (hm. 5.11) ee dditional nswers. ection 5.6 hscc_geo_te_0506.indd /1/15 3:04 M

63 NW ee dditional nswers. 31. ample answer: Mini-ssessment 3. yes; oth triangles will always have 3 vertices. o, each vertex of one triangle can be mapped to one vertex of the other triangle so that each vertex is only used once, and adjacent vertices of one triangle must be adjacent in the other ee dditional nswers. an the triangles be proven congruent with the information given in the diagram? If so, state the theorem you would use. 1. M MN M N Yes, hm or hm MOLING WIH MHMI When a light ray from an object meets a mirror, it is reflected back to your eye. or example, in the diagram, a light ray from point is reflected at point and travels back to point. he law of reflection states that the angle of incidence,, is congruent to the angle of reflection,. a. rove that is congruent to. Given, rove b. Verify that is isosceles. c. oes moving away from the mirror have any effect on the amount of his or her reflection a person sees? xplain. 30. HOW O YOU I? Name as many pairs of congruent triangles as you can from the diagram. xplain how you know that each pair of triangles is congruent. Maintaining Mathematical roficiency 31. ONUION onstruct a triangle. how that there is no congruence rule by constructing a second triangle that has the same angle measures but is not congruent. 3. HOUGH OVOKING Graph theory is a branch of mathematics that studies vertices and the way they are connected. In graph theory, two polygons are isomorphic if there is a one-to-one mapping from one polygon s vertices to the other polygon s vertices that preserves adjacent vertices. In graph theory, are any two triangles isomorphic? xplain your reasoning. 33. MHMIL ONNION ix statements are given about UV and XYZ. U XY UV YZ V XZ X U Y V Z U V Z X a. List all combinations of three given statements that would provide enough information to prove that UV is congruent to XYZ. b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent? ind the coordinates of the midpoint of the line segment with the given endpoints. (ection 1.3) 34. (1, 0) and (5, 4) 35. J(, 3) and K(4, 1) 36. ( 5, 7) and (, 4) opy the angle using a compass and straightedge. (ection 1.5) eviewing what you learned in previous grades and lessons Y 76 hapter 5 ongruent riangles No 3. If students need help... esources by hapter ractice and ractice uzzle ime If students got it... esources by hapter nrichment and xtension umulative eview tudent Journal ractice tart the next ection Yes, hm or hm ifferentiating the Lesson kills eview Handbook 76 hapter 5

64 Laurie s Notes Overview of ection 5.7 Introduction his is a short lesson that extends students understanding of congruent triangles. Once two triangles have been proven congruent using three corresponding parts in the two triangles, there are additional corresponding parts that can now be deduced to be congruent. In addition to using corresponding parts of congruent triangles, prior compass constructions will also be proven as valid methods. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. acing uggestion If both explorations have been explored thoroughly, including the proofs, you may feel comfortable beginning the formal lesson with xample 3. ection

65 ommon ore tate tandards HG-..5 Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures. Laurie s Notes xploration Motivate sk students what they know about Napoleon onaparte. Here are a few facts that you might expect to hear: military leader and first emperor of rance lived considered one of the world s greatest military leaders led reform within rance (legal system, education, economy, and church) xplain to students that the second exploration is connected to Napoleon. iscuss If you know your community of students and families, it is possible that someone has access to an electronic transit. If so, perhaps they would come to your class to demonstrate how the transit works and to do an indirect measurement of some object or distance near your school. You can also simulate the method in your classroom using rope or yarn. ven though you can reach all of the areas in your room, pretend there is a gorge that cannot be crossed but could be sited with the transit! You want to find the distance across the gorge. Use the method from xploration 1 or. xploration 1 iscuss with students what a transit is and what a surveyor does. Have partners study the picture. sk at least two volunteers to share their proof at the board. lthough the method will be the same (), the order in which students write the steps may well vary. M3 onstruct Viable rguments and ritique the easoning of Others: In examining peers proofs, students should critique the logical progression of statements and whether the reasoning is correct. xploration his problem is similar to the first exploration and again uses corresponding parts of congruent triangles to determine a distance. You could replicate this method outside on a playing field or in the school s gymnasium. ommunicate Your nswer Have students share their descriptions of indirect measurement. onnecting to Next tep he explorations give students time to explore in real-life problems how corresponding parts of congruent triangles can be used. Once explored, students will be better able to consider deductive proofs using congruent triangles. -77 hapter 5

66 IIUING H ONING O OH 5.7 o be proficient in math, you need to listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Using ongruent riangles ssential uestion How can you use congruent triangles to make an indirect measurement? Work with a partner. he figure shows how a surveyor can measure the width of a river by making measurements on only one side of the river. a. tudy the figure. hen explain how the surveyor can find the width of the river. Measuring the Width of a iver b. Write a proof to verify that the method you described in part (a) is valid. Given is a right angle, is a right angle, c. xchange proofs with your partner and discuss the reasoning used. Measuring the Width of a iver Work with a partner. It was reported that one of Napoleon s officers estimated the width of a river as follows. he officer stood on the bank of the river and lowered the visor on his cap until the farthest thing visible was the edge of the bank on the other side. He then turned and noted the point on his side that was in line with the tip of his visor and his eye. he officer then paced the distance to this point and concluded that distance was the width of the river. a. tudy the figure. hen explain how the officer concluded that the width of the river is G. b. Write a proof to verify that the conclusion the officer made is correct. Given G is a right angle, is a right angle, G c. xchange proofs with your partner and discuss the reasoning used. ommunicate Your nswer 3. How can you use congruent triangles to make an indirect measurement? 4. Why do you think the types of measurements described in xplorations 1 and are called indirect measurements? G ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. ee dditional nswers. 3. y creating a triangle that is congruent to a triangle with an unknown side length or angle measure, you can measure the created triangle and use it to find the unknown measure indirectly. 4. You do not actually measure the side length or angle measure you are trying to find. You measure the side length or angle measure of a triangle that is congruent to the one you are trying to find. ection 5.7 Using ongruent riangles 77 ection

67 ifferentiated Instruction Visual When working with overlapping triangles, students may find it helpful to redraw the figure as two separate figures, and then use colors to designate corresponding parts. xtra xample 1 Use the diagram in xample 1. xplain how using this new given information would change the proof in xample 1. Given, rove Instead of the hm. (5.11), the proof would use the hm. (5.10) to prove the triangles congruent. MONIOING OG NW 1. ll three pairs of sides are congruent. o, by the ongruence heorem (hm. 5.8),. ecause corresponding parts of congruent triangles are congruent,. 5.7 Lesson ore Vocabulary revious congruent figures corresponding parts construction What You Will Learn Use congruent triangles. rove constructions. Using ongruent riangles ongruent triangles have congruent corresponding parts. o, if you can prove that two triangles are congruent, then you know that their corresponding parts must be congruent as well. Using ongruent riangles xplain how you can use the given information to prove that the hang glider parts are congruent. 1 Given 1, rove OLUION If you can show that, then you will know that. irst, copy the diagram and mark the given information. hen mark the information that you can deduce. In this case, and are supplementary to congruent angles, so. lso, by the eflexive roperty of ongruence (heorem.1). Mark given information. Mark deduced information. 1 wo angle pairs and a non-included side are congruent, so by the ongruence heorem (heorem 5.11),. ecause corresponding parts of congruent triangles are congruent,. Monitoring rogress Help in nglish and panish at igideasmath.com 1. xplain how you can prove that. 78 hapter 5 ongruent riangles Laurie s Notes eacher ctions ig Idea: It takes three pieces of information to prove that two triangles are congruent (,,,, HL). Once proven congruent, there are three additional pieces of information that can be deduced to be true about the two triangles. xplain to students that in this lesson they will prove triangles congruent. ny corresponding parts that were not used in the proof can now be deduced to be congruent. iscuss xample 1. Have partners work together to write the proof. 78 hapter 5

68 MKING N O OLM When you cannot easily measure a length directly, you can make conclusions about the length indirectly, usually by calculations based on known lengths. Use the following method to find the distance across a river, from point N to point. lace a stake at K on the near side so that NK N. ind M, the midpoint of NK. Locate the point L so that NK KL and L,, and M are collinear. xplain how this plan allows you to find the distance. Using ongruent riangles for Measurement L N M K xtra xample xplain how to use the measurements in the diagram to find the distance across the pond. 55 yd 68 yd 57 yd 68 yd 55 yd OLUION ecause NK N and NK KL, N and K are congruent right angles. ecause M is the midpoint of NK, NM KM. he vertical angles KML and NM are congruent. o, MLK MN by the ongruence heorem (heorem 5.10). hen because corresponding parts of congruent triangles are congruent, KL N. o, you can find the distance N across the river by measuring KL. L N M K y Vertical s hm. (.6),. ecause = 55, = 55, = 68, and = 68, and. y hm. (5.5),. o, by, and the length of the pond is 57 yards. lanning a roof Involving airs of riangles Use the given information to write a plan for proof. Given 1, 3 4 rove OLUION In and, you know that 1 and. If you can show that, then you can use the ongruence heorem (heorem 5.5). o prove that, you can first prove that. You are given 1 and 3 4. by the eflexive roperty of ongruence (heorem.1). You can use the ongruence heorem (heorem 5.10) to prove that. lan for roof Use the ongruence heorem (heorem 5.10) to prove that. hen state that. Use the ongruence heorem (heorem 5.5) to prove that. Monitoring rogress Help in nglish and panish at igideasmath.com. In xample, does it matter how far from point N you place a stake at point K? xplain. 3. Write a plan to prove that U U. ection 5.7 Using ongruent riangles U xtra xample 3 Use the diagram in xample 3. Write a plan to prove that. lan for roof Use the ongruence heorem (hm. 5.10) to prove that. hen state that. Use the ongruence heorem (hm. 5.5) to prove that. MONIOING OG NW. no; s long as the rest of the steps are followed correctly, LKM will still be congruent to NM. o, the corresponding parts will still be congruent, and you will still be able to find N by measuring LK. 3. Use the HL ongruence heorem (hm. 5.9) to prove that U. Use the ongruence heorem (hm. 5.5) to prove that U. hen state that U. Use the ongruence heorem (hm. 5.8) to prove that U U. Laurie s Notes eacher ctions Have students sketch the diagram for xample. s you read the method for constructing the triangles, have students mark the diagram with the known information. Is there additional information that you can deduce? yes; Vertical angles are congruent. M1 Make ense of roblems and ersevere in olving hem: iscuss the plan for the proof in xample 3. Have students self-assess with humbs Up. Have partners work together to write the proof. esponse Logs: It helped me today when ection

69 xtra xample 4 Write a proof to verify that the construction of the midpoint of a segment is valid. M Given rove M is the midpoint of. tatements (easons) 1. (Given). (efl. rop. hm..1) 3. ( hm. 5.8) 4. M M () 5. M M (efl. rop. hm..1) 6. M M ( hm. 5.5) 7. M M () 8. M is the midpoint of. (ef. of midpoint) MONIOING OG NW 4. and roving onstructions ecall that you can use a compass and a straightedge to copy an angle. he construction is shown below. You can use congruent triangles to prove that this construction is valid. tep 1 tep tep 3 raw a segment and arcs o copy, draw a segment with initial point. raw an arc with center. Using the same radius, draw an arc with center. Label points,, and. raw an arc raw an arc with radius and center. Label the intersection. roving a onstruction raw a ray raw. In xample 4, you will prove that. Write a proof to verify that the construction for copying an angle is valid. OLUION dd and to the diagram. In the construction, one compass setting determines,,, and, and another compass setting determines and. o, you can assume the following as given statements. Given,, rove lan for roof lan in ction how that, so you can conclude that the corresponding parts and are congruent. MN 1.,, ON 1. Given.. ongruence heorem (heorem 5.8) orresponding parts of congruent triangles are congruent. Monitoring rogress Help in nglish and panish at igideasmath.com 4. Use the construction of an angle bisector on page 4. What segments can you assume are congruent? 80 hapter 5 ongruent riangles Laurie s Notes eacher ctions xplain that the constructions students have done in this chapter can be proven valid by using one of the triangle-congruence theorems. ifferentiate and have different groups of students prove one of the constructions presented in this chapter. Have each group present their proof to the class. losure esponse Logs: elect from I didn t expect or I learned to or he hardest part right now is. 80 hapter 5

70 5.7 xercises In xercises 3 8, explain how to prove that the statement is true. (ee xample 1.) J J 5. JM LM 6. K M 7. GK HJ 8. W V G N M L H L K In xercises 9 1, write a plan to prove that 1. (ee xample 3.) K 1 J Vocabulary and ore oncept heck 1. OML H NN parts of congruent triangles are congruent.. WIING escribe a situation in which you might choose to use indirect measurement with congruent triangles to find a measure rather than measuring directly. Monitoring rogress and Modeling with Mathematics 1 G H V U W rom the diagram, W V and VW. o, by the ongruence heorem (hm. 5.11), VW V. ecause corresponding parts of congruent triangles are congruent, W V. 9. Use the ongruence heorem (hm. 5.11) to prove that HG GK. hen, state that GK GH. Use the ongruent omplements heorem (hm..5) to prove that Use the ongruence heorem (hm. 5.11) to prove that. hen, ynamic olutions available at igideasmath.com In xercises 13 and 14, write a proof to verify that the construction is valid. (ee xample 4.) 13. Line perpendicular to a line through a point not on the line lan for roof how that by the ongruence heorem (heorem 5.8). hen show that M M using the ongruence heorem (heorem 5.5). Use corresponding parts of congruent triangles to show that M and M are right angles. 14. Line perpendicular to a line through a point on the line lan for roof how that by the ongruence heorem (heorem 5.8). Use corresponding parts of congruent triangles to show that and are right angles. In xercises 15 and 16, use the information given in the diagram to write a proof. 15. rove L HN H G J ection 5.7 Using ongruent riangles 81 L M state that because corresponding parts of congruent triangles are congruent. Use the ase ngles heorem (hm. 5.6) to prove that Use the ongruence heorem (hm. 5.10) to prove that. hen, state that because corresponding parts of congruent triangles are congruent. Use the ongruence heorem (hm. 5.5) to prove that. o, ee dditional nswers. M K N ssignment Guide and Homework heck IGNMN asic: 1,, 3 17 odd, 18, 1, 4, 5 verage: 1,, 6 18 even, 1,, 4, 5 dvanced: 1,, 6 0 even, 1 5 HOMWOK HK asic: 3, 11, 13, 15, 18 verage: 6, 10, 14, 16, 18 dvanced: 10, 14, 16, 18, NW 1. orresponding. ample answer: You might use indirect measurement to find the height of a cave opening on the side of a cliff. 3. ll three pairs of sides are congruent. o, by the ongruence heorem (hm. 5.8),. ecause corresponding parts of congruent triangles are congruent,. 4. wo pairs of sides and the pair of included angles are congruent. o, by the ongruence heorem (hm. 5.5),. ecause corresponding parts of congruent triangles are congruent,. 5. he hypotenuses and one pair of legs of two right triangles are congruent. o, by the HL ongruence heorem (hm. 5.9), JMK LMK. ecause corresponding parts of congruent triangles are congruent, JM LM. 6. by the lternate Interior ngles heorem (hm. 3.). rom the diagram,. by the eflexive roperty of ongruence (hm..1). o, by the ongruence heorem (hm. 5.11),. ecause corresponding parts of congruent triangles are congruent,. 7. rom the diagram, JHN KGL, N L, and JN KL. o, by the ongruence heorem (hm. 5.11), JNH KLG. ecause corresponding parts of congruent triangles are congruent, GK HJ. ection

71 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW ee dditional nswers. 18. a. the red triangle b. 4 times 19. ee dditional nswers. 0. M 1034 mi, 956 mi, M 1035 mi, 305 mi, 439,000 mi ; ample answer: triangle with vertices at opeka, K, uffalo, NY, and Orlando, L is approximately congruent to the ermuda riangle ee dditional nswers. Mini-ssessment ecide whether the statement must be true. xplain your reasoning. J G M H K 16. rove UX Y X 17. MOLING WIH MHMI xplain how to find the distance across the canyon. (ee xample.) 18. HOW O YOU I? Use the tangram puzzle. a. Which triangle(s) have an area that is twice the area of the purple triangle? W b. How many times greater is the area of the orange triangle than the area of the purple triangle? 19. OO rove that the green triangles in the Jamaican flag are congruent if and is the midpoint of. U Y V 0. HOUGH OVOKING he ermuda riangle is a region in the tlantic Ocean in which many ships and planes have mysteriously disappeared. he vertices are Miami, an Juan, and ermuda. Use the Internet or some other resource to find the side lengths, the perimeter, and the area of this triangle (in miles). hen create a congruent triangle on land using cities as vertices. Miami, L ermuda an Juan, uerto ico 1. MKING N GUMN Your friend claims that WZY can be proven congruent to YXW using the HL ongruence W heorem (hm. 5.9). Is your friend correct? xplain your reasoning. Z. IIL HINKING etermine whether each conditional statement is true or false. If the statement is false, rewrite it as a true statement using the converse, inverse, or contrapositive. a. If two triangles have the same perimeter, then they are congruent. b. If two triangles are congruent, then they have the same area. 3. NING O IION Which triangles are congruent to? elect all that apply. M X Y L 1. L No. ample answer: In KL you don t know that G + GK = LM + MK. o, KL is not isosceles and its base angles and L are not congruent.. L N Yes. JML KMN by hm. (5.5). o, by, L N. 3. Write a plan to prove that U V. X W U N lan for roof how that U V because = UV so + U = U + UV. It is given that U V and U V. o, by hm. (5.10), U V. Y V 8 hapter 5 ongruent riangles Maintaining Mathematical roficiency ind the perimeter of the polygon with the given vertices. (ection 1.4) 4. ( 1, 1), (4, 1), (4, ), ( 1, ) 5. J( 5, 3), K(, 1), L(3, 4) If students need help... esources by hapter ractice and ractice uzzle ime tudent Journal ractice ifferentiating the Lesson kills eview Handbook J G eviewing what you learned in previous grades and lessons If students got it... esources by hapter nrichment and xtension umulative eview tart the next ection H K N L 8 hapter 5

72 Laurie s Notes Overview of ection 5.8 Introduction his lesson presents a new type of proof format, the coordinate proof. While students have used formulas to find distances, slopes, and midpoints, in this lesson the coordinates are replaced with variables. When a proof is done with variable coordinates, the results are true for all figures of that type. tudents will recognize that the orientation of a figure in the coordinate plane can assist in making the computations easier. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations ormative ssessment ips efer to page -30 for a description of esponse Logs. In this lesson and throughout this chapter, there will be sentence stems to prompt student writing. Use your judgment for placement in the lesson, amount of time to allow for writing, and the actual sentence stem given to students. or a beginning list of writing stems, see page -38. nother Way he two explorations could be done with pencil and graph paper because there is no need to make this a dynamic sketch. acing uggestion Once students have completed the explorations, continue with the formal lesson. ection 5.8-8

73 ommon ore tate tandards HG-G..4 Use coordinates to prove simple geometric theorems algebraically. Laurie s Notes xploration Motivate Write on the board: ( 3, 0), (, 3), and (5, ). ivide the class into three groups:,, and. Have each group find the slope, length, 3 and midpoint of their segment. : 5, 34, (, 3 ); : 5 3, 34, ( 7, 1 ); : 1 4, 68, (1, 1) Have a volunteer from each group record the results of their work. urn and alk: What type of triangle is? How do you know? he triangle is a right isosceles triangle. xplanations will vary. ome may use slope to justify that there is a right angle. Others may use the onverse of the ythagorean heorem (grade 8). he istance ormula shows that the triangle is isosceles. xplain to students that they are going to work with coordinate proofs in this lesson, and the formulas learned in prior math classes will be important. 1 xploration hese explorations as well as the examples in the formal lesson require that students recall the istance ormula, Midpoint ormula, and slope formula. xploration 1 You know the coordinates of points,, and. What is necessary to prove that is isosceles? You need at least two congruent sides. Without using the measurement tool, how can you show that there are at least two congruent sides? Use the istance ormula to find and. onfirm that students recall the istance ormula. xploration What is necessary to prove that is a right isosceles triangle? how that two sides are congruent and that the angle between the congruent sides is a right angle. How do you show that the sides form a right angle? how that the sides are perpendicular by finding the slope of each line segment. Have partners work through the algebra to show that is a right isosceles triangle. Have different groups of students select a different point for below the x-axis, such as (3, 1), (3, ), (3, 3), or (3, 4). M3 onstruct Viable rguments and ritique the easoning of Others: Have students summarize what they believe is true in this exploration. Listen for an understanding that as long as is on the line x = 3, is isosceles. If is (3, 3) or (3, 3), then the triangle is also a right triangle. xtension: Have students also explore when will be an obtuse isosceles triangle and when it will be an acute isosceles triangle. ommunicate Your nswer M1 Make ense of roblems and ersevere in olving hem and M eason bstractly and uantitatively: kills learned in an algebra class are essential in writing a coordinate proof. Using variables as the coordinates versus real numbers means that the proof is true for all figures of this type. onnecting to Next tep he explorations give students the opportunity to recall and use the algebraic techniques that will be necessary in writing coordinate proofs in the formal lesson. -83 hapter 5

74 IIUING H ONING O OH 5.8 o be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results. oordinate roofs ssential uestion How can you use a coordinate plane to write a proof? Work with a partner. a. Use dynamic geometry software to draw with endpoints (0, 0) and (6, 0). b. raw the vertical line x = 3. c. raw so that lies on the line x = 3. d. Use your drawing to prove that is an isosceles triangle. Writing a oordinate roof Writing a oordinate roof Work with a partner. a. Use dynamic geometry software to draw with endpoints (0, 0) and (6, 0). b. raw the vertical line x = 3. c. lot the point (3, 3) and draw. hen use your drawing to prove that is an isosceles right triangle ample oints (0, 0) (6, 0) (3, 3) egments = 6 = 4.4 = 4.4 Line x = 3 d. hange the coordinates of so that lies below the x-axis and is an isosceles right triangle. e. Write a coordinate proof to show that if lies on the line x = 3 and is an isosceles right triangle, then must be the point (3, 3) or the point found in part (d). ommunicate Your nswer 3. You can position the figure in a coordinate plane and then use deductive reasoning to show that what you are trying to prove must be true based on the coordinates of the figure. 4. Using the istance ormula, = 6, = 6, and = 6., so is an equilateral triangle How can you use a coordinate plane to write a proof? 4. Write a coordinate proof to prove that with vertices (0, 0), (6, 0), and (3, 3 3 ) is an equilateral triangle ection 5.8 oordinate roofs 83 ample oints (0, 0) (6, 0) (3, y) egments = 6 Line x = 3 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Lesson lanning ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. a. heck students work. b. heck students work. c. heck students work. d. Using the istance ormula, = 9 + y and = 9 + y., so is an isosceles triangle.. a. heck students work. b. heck students work. c. heck students work; = 3, m = 1, = 3, and m = 1. o, and is a right isosceles triangle. d. (3, 3) y x 90 e. If lies on the line x = 3, then the coordinates are (3, y). ecause is an isosceles triangle, m = y 3, and m = y 3. is a right triangle, so it must have a right angle. ecause and are the congruent legs of, and are the congruent base angles by the ase ngles heorem (hm. 5.6). he vertex angle,, must be the right angle, which means that by the definition of perpendicular lines. y the lopes of erpendicular Lines heorem (hm. 3.14), y 3 y = 1 and y = ±3. 3 o, the coordinates of must be (3, 3) or (3, 3). ection

75 nglish Language Learners Group ctivity orm small groups of nglish learners and nglish speakers. Have students discuss and illustrate different ways to place a square in a coordinate plane. licit suggestions for different ways to label the vertices so that it is convenient to find the lengths of the sides of the square. xtra xample 1 lace each figure in a coordinate plane in a way that is convenient for finding side lengths. ssign coordinates to each vertex. a. scalene right triangle (0, b) y 5.8 Lesson ore Vocabulary coordinate proof, p. 84 What You Will Learn lace figures in a coordinate plane. Write coordinate proofs. lacing igures in a oordinate lane coordinate proof involves placing geometric figures in a coordinate plane. When you use variables to represent the coordinates of a figure in a coordinate proof, the results are true for all figures of that type. lacing a igure in a oordinate lane lace each figure in a coordinate plane in a way that is convenient for finding side lengths. ssign coordinates to each vertex. a. a rectangle b. a scalene triangle OLUION It is easy to find lengths of horizontal and vertical segments and distances from (0, 0), so place one vertex at the origin and one or more sides on an axis. a. Let h represent the length and b. Notice that you need to use k represent the width. three different variables. y (0, k) k (h, k) (0, 0) h (h, 0) x y (f, g) (0, 0) (d, 0) x (0, 0) b. isosceles trapezoid y (a, 0) x Monitoring rogress Help in nglish and panish at igideasmath.com 1. how another way to place the rectangle in xample 1 part (a) that is convenient for finding side lengths. ssign new coordinates.. square has vertices (0, 0), (m, 0), and (0, m). ind the fourth vertex. ( a, b) (a, b) Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure. Writing a lan for a oordinate roof ( c, 0) (c, 0) x Write a plan to prove that O bisects. Given oordinates of vertices of O and O rove O bisects. y (0, 4) (3, 0) xtra xample Use the information in xample. Write a plan for a proof that uses the ongruence heorem (hm. 5.5). lan for roof he x- and y-axes are perpendicular, so use definitions to show O O. Use the istance ormula to show O = O. Use the eflexive rop. (hm..1) to show O O. Use hm. (5.5) to show that O O. hen use to conclude that O O, which implies O bisects. MONIOING OG NW 1. ee dditional nswers.. (m, m) 84 hapter 5 ongruent riangles Laurie s Notes ( 3, 0) O(0, 0) 4 x OLUION lan for roof Use the istance ormula to find the side lengths of O and O. hen use the ongruence heorem (heorem 5.8) to show that O O. inally, use the fact that corresponding parts of congruent triangles are congruent to conclude that O O, which implies that O bisects. eacher ctions xplain the difference between performing a calculation using real numbers and using variables. efine a coordinate proof. raw coordinate axes without scaling either axis. Locate a point at the origin, on the positive x-axis, and in the first quadrant. What coordinates do you know with certainty? (0, 0) and the y-coordinate of the point on the positive x-axis urn and alk: In xample, how could you show that O bisects? tudents will see that the triangles can be proven congruent. 84 hapter 5

76 INING N NY OIN nother way to solve xample 3 is to place a triangle with point at (0, h) on the y-axis and hypotenuse on the x-axis. o make a right angle, position and so that legs and have slopes of 1 and 1, respectively. lope is 1. y lope is 1. (0, h) ( h, 0) (h, 0) x Length of hypotenuse = h M ( h + h, ) = M(0, 0) Monitoring rogress 3. Write a plan for the proof. Given GJ bisects OGH. rove GJO GJH he coordinate proof in xample applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates. or instance, you can use variable coordinates to duplicate the proof in xample. Once this is done, you can conclude that O bisects for any triangle whose coordinates fit the given pattern. Help in nglish and panish at igideasmath.com 6 4 O y J G ( h, 0) pplying Variable oordinates 4 H x y (0, k) (h, 0) O(0, 0) x lace an isosceles right triangle in a coordinate plane. hen find the length of the hypotenuse and the coordinates of its midpoint M. OLUION lace O with the right angle at the origin. Let the length of the legs be k. hen the vertices are located at (0, k), (k, 0), and O(0, 0). (0, k) y M O(0, 0) (k, 0) x Use the istance ormula to find, the length of the hypotenuse. = (k 0) + (0 k) = k + ( k) = k + k = k = k Use the Midpoint ormula to find the midpoint M of the hypotenuse. M ( 0 + k, k + 0 ) ( = M k, k ) o, the length of the hypotenuse is k and the midpoint of the hypotenuse is ( k, k ). Monitoring rogress Help in nglish and panish at igideasmath.com 4. Graph the points O(0, 0), H(m, n), and J(m, 0). Is OHJ a right triangle? ind the side lengths and the coordinates of the midpoint of each side. xtra xample 3 lace an isosceles triangle on a coordinate plane with vertices ( a, 0), (0, a), and (a, 0). ind the length of each side. hen find the coordinates of the midpoint of each side. = = a 5, = 4a; the midpoint of is ( a, a ) the midpoint of is ( a, a midpoint of is (0, 0)., and the ), MONIOING OG NW 3. Use the istance ormula to find GO and GH in order to show that GO GH. tate that OGJ HGJ by the definition of angle bisector and GJ GJ by the eflexive roperty of ongruence (hm..1). hen use the ongruence heorem (hm. 5.5) to show that GJO GJH. 4. y H(m, n) O(0, 0) J(m, 0) yes; OJ = m, M OJ ( m, 0 ), HJ = n, M HJ ( m, ) n, x ) OH = m + n, M OH ( m, n ection 5.8 oordinate roofs 85 Laurie s Notes eacher ctions In xample, how would the proof change if the coordinates of and were ( a, 0) and (a, 0)? he process and proof would be the same. iscuss elements of a coordinate proof. ositioning a polygon to make use of the origin and intercepts is common. M1: ose xample 3, noting that there are two suggested orientations. his problem is related to a later lesson when students will determine that the length of the hypotenuse of a right isosceles triangle is times the length of the leg. ection

77 xtra xample 4 Write a coordinate proof. Use the diagram and the given information in xample 4. rove that OU VUO. he slope of O is k 0 m 0 = k. he slope m of UV k 0 is (m + h) h = k m. o, O UV. y the lternate Interior ngles heorem (hm. 3.), OU VUO. xtra xample 5 s part of a graphic design, you draw a rectangle and then connect the midpoints of the sides. rove that the quadrilateral MN formed has four congruent sides. y (0, a) N (b, a) M (0, 0) (b, 0) he coordinates of the midpoints are M(0, a), N(b, a), (b, a), and (b, 0). he lengths of MN, N,, and M are (b 0) + (a a), (b b) + (a a), (b b) + (a 0), and (b 0) + (0 a). ach expression simplifies to b + a, so the four segments are congruent. x Writing oordinate roofs Write a coordinate proof. Given oordinates of vertices of quadrilateral OUV rove OU UVO Writing a oordinate roof OLUION egments OV and U have the same length. OV = h 0 = h U = (m + h) m = h O(0, 0) y (m, k) U(m + h, k) V(h, 0) Horizontal segments U and OV each have a slope of 0, which implies that they are parallel. egment OU intersects U and OV to form congruent alternate interior angles, UO and VOU. y the eflexive roperty of ongruence (heorem.1), OU OU. o, you can apply the ongruence heorem (heorem 5.5) to conclude that OU UVO. Writing a oordinate roof You buy a tall, three-legged plant stand. When you place a plant on the stand, the stand appears to be unstable under the weight of the plant. he diagram at the right shows a coordinate plane superimposed on one pair of the plant stand s legs. he legs are extended to form O. rove that O is a scalene triangle. xplain why the plant stand may be unstable. OLUION irst, find the side lengths of O. O = (48 0) + (1 0) = = (18 1) + (0 48) = O = 18 0 = 18 y 48 ecause O has no congruent sides, O is a scalene triangle by definition. he plant stand may be unstable because O is longer than, so the plant stand is leaning to the right (1, 48) O(0, 0) 1 (18, 0) x x MONIOING OG NW 5. Using the istance ormula, N = h, MN = h, O = h, and MO = h. o, N MN and O MO, and by the eflexive roperty of ongruence (hm..1), ON ON. o, you can apply the ongruence heorem (hm. 5.8) to conclude that NO NMO. 86 hapter 5 ongruent riangles Monitoring rogress 5. Write a coordinate proof. Given oordinates of vertices of NO and NMO rove NO NMO Help in nglish and panish at igideasmath.com y (0, h) N(h, h) O(0, 0) M(h, 0) x Laurie s Notes eacher ctions In xample 4, help students make sense of the coordinates for vertex U. How can you show that two segments are parallel? tudents may say they need alternate interior angles congruent or show that the line segments have the same slope. hink-air-hare: Have students try xample 5 independently before discussing as a class. losure esponse Logs: elect from I figured these out because or I am feeling good about or When I am writing a proof. 86 hapter 5

78 5.8 xercises Vocabulary and ore oncept heck 1. VOULY How is a coordinate proof different from other types of proofs you have studied? How is it the same?. WIING xplain why it is convenient to place a right triangle on the grid as shown when writing a coordinate proof. ynamic olutions available at igideasmath.com (0, b) y ssignment Guide and Homework heck IGNMN asic: 1,, 3 19 odd, 0, 6, 8, 9 verage: 1,, 6 even, 6, 8, 9 dvanced: 1,, 6, 1 0 even, 9 (0, 0) Monitoring rogress and Modeling with Mathematics (a, 0) x HOMWOK HK asic: 5, 7, 13, 15, 17 verage: 6, 1, 14, 16, 18 dvanced: 6, 1, 14, 18, In xercises 3 6, place the figure in a coordinate plane in a convenient way. ssign coordinates to each vertex. xplain the advantages of your placement. (ee xample 1.) 3. a right triangle with leg lengths of 3 units and units 4. a square with a side length of 3 units 5. an isosceles right triangle with leg length p 6. a scalene triangle with one side length of m In xercises 7 and 8, write a plan for the proof. (ee xample.) 7. Given oordinates of vertices of OM and ONM rove OM and ONM are isosceles triangles. y (3, 4) 4 M(8, 4) O(0, 0) N(5, 0) 8 x 8. Given G is the midpoint of H. rove GHJ GO y H(1, 4) J(6, 4) 4 G O(0, 0) 4 (5, 0) x 5. ample answer: (0, p) (0, 0) (p, 0) x It is easy to find the lengths of horizontal and vertical segments and distances from the origin. y In xercises 9 1, place the figure in a coordinate plane and find the indicated length. 9. a right triangle with leg lengths of 7 and 9 units; ind the length of the hypotenuse. 10. an isosceles triangle with a base length of 60 units and a height of 50 units; ind the length of one of the legs. 11. a rectangle with a length of 5 units and a width of 4 units; ind the length of the diagonal. 1. a square with side length n; ind the length of the diagonal. In xercises 13 and 14, graph the triangle with the given vertices. ind the length and the slope of each side of the triangle. hen find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? xplain. (ssume all variables are positive and m n.) (ee xample 3.) 13. (0, 0), (h, h), (h, 0) 14. (0, n), (m, n), (m, 0) In xercises 15 and 16, find the coordinates of any unlabeled vertices. hen find the indicated length(s). 15. ind ON and MN. 16. ind O. y N k units O(0, 0) (h, 0) M(h, 0) x O ection 5.8 oordinate roofs ample answer: J(0, 0) y y L(n, p) U K(m, 0) x k units It is easy to find the lengths of horizontal segments and distances from the origin ee dditional nswers. x NW 1. In a coordinate proof, you have to assign coordinates to vertices and write expressions for the side lengths and the slopes of segments in order to show how sides are related; s with other types of proofs, you still have to use deductive reasoning and justify every conclusion with theorems, proofs, and properties of mathematics.. When the triangle is positioned as shown, you are using zeros in your expressions, so the side lengths are often the same as one of the coordinates. 3. ample answer: 3 1 (0, 0) y (0, ) 1 3 (3, 0) It is easy to find the lengths of horizontal and vertical segments and distances from the origin. 4. ample answer: y W(0, 3) Z(3, 3) 3 1 X(0, 0) 1 x Y(3, 0) 3 x It is easy to find the lengths of horizontal and vertical segments and distances from the origin. ection

79 ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW ee dditional nswers ee dditional nswers. Mini-ssessment upply the missing coordinates. 1. O is a square. y (0, a) (?,?) O(0, 0) (a, a), (a, 0). OG is isosceles. y (a, b) O(0, 0) x (?,?) x G(?,?) G(a, 0) 3. Write a coordinate proof. Given oordinates of the vertices of isosceles trapezoid HIJK, M is the midpoint of HK, N is the midpoint of IJ. rove MN = 1 ( HI + KJ) y OO In xercises 17 and 18, write a coordinate proof. (ee xample 4.) 17. Given oordinates of vertices of and O rove O y (h, k) (h, k) O(0, 0) (h, 0) (h, k) 18. Given oordinates of, H is the midpoint of, G is the midpoint of. rove G H y (0, k) O (h, 0) x 88 hapter 5 ongruent riangles H ( h, 0) 19. MOLING WIH MHMI You and your cousin are camping in the woods. You hike to a point that is 500 meters east and 100 meters north of the campsite. Your cousin hikes to a point that is 1000 meters east of the campsite. Use a coordinate proof to prove that the triangle formed by your position, your cousin s position, and the campsite is isosceles. (ee xample 5.) 0. MKING N GUMN wo friends see a drawing of quadrilateral with vertices (0, ), (3, 4), (1, 5), and (, 1). One friend says the quadrilateral is a parallelogram but not a rectangle. he other friend says the quadrilateral is a rectangle. Which friend is correct? Use a coordinate proof to support your answer. 1. MHMIL ONNION Write an algebraic expression for the coordinates of each endpoint of a line segment whose midpoint is the origin. Maintaining Mathematical roficiency G x. ONING he vertices of a parallelogram are (w, 0), (0, v), ( w, 0), and (0, v). What is the midpoint of the side in uadrant III? ( w, v ) ( w, ) v ( w, v ) ( w, ) v 3. ONING rectangle with a length of 3h and a width of k has a vertex at ( h, k). Which point cannot be a vertex of the rectangle? (h, k) ( h, 0) (h, 0) (h, k) 4. HOUGH OVOKING hoose one of the theorems you have encountered up to this point that you think would be easier to prove with a coordinate proof than with another type of proof. xplain your reasoning. hen write a coordinate proof. 5. IIL HINKING he coordinates of a triangle are (5d, 5d ), (0, 5d ), and (5d, 0). How should the coordinates be changed to make a coordinate proof easier to complete? 6. HOW O YOU I? Without performing any calculations, how do you know that the diagonals of square UVW are perpendicular to each other? How can you use a similar diagram to show that the diagonals of any square are perpendicular to each other? W(, 0) 4 y 3 (0, ) U(, 0) 4 V(0, ) 3 7. OO Write a coordinate proof for each statement. a. he midpoint of the hypotenuse of a right triangle is the same distance from each vertex of the triangle. b. ny two congruent right isosceles triangles can be combined to form a single isosceles triangle. YW bisects XYZ such that m XYW = (3x 7) and m WYZ = (x + 1). (ection 1.5) 8. ind the value of x. 9. ind m XYZ. eviewing what you learned in previous grades and lessons x H( a, b) I(a, b) M K( c, 0) N x J(c, 0) irst, find the coordinates of M and N: M( a c, b), N(a + c, b). hen find that MN = a + c, HI = 4a, and JK = 4c. o, 1 ( HI + KJ) = a + c = MN. If students need help... esources by hapter ractice and ractice uzzle ime tudent Journal ractice ifferentiating the Lesson kills eview Handbook If students got it... esources by hapter nrichment and xtension umulative eview tart the next ection 88 hapter 5

80 What id You Learn? ore Vocabulary legs (of a right triangle), p. 64 hypotenuse (of a right triangle), p. 64 coordinate proof, p. 84 ore oncepts heorem 5.8 ide-ide-ide () ongruence heorem, p. 6 heorem 5.9 Hypotenuse-Leg (HL) ongruence heorem, p. 64 heorem 5.10 ngle-ide-ngle () ongruence heorem, p. 70 heorem 5.11 ngle-ngle-ide () ongruence heorem, p. 71 Using ongruent riangles, p. 78 roving onstructions, p. 80 lacing igures in a oordinate lane, p. 84 Writing oordinate roofs, p. 86 Mathematical ractices 1. Write a simpler problem that is similar to xercise on page 67. escribe how to use the simpler problem to gain insight into the solution of the more complicated problem in xercise.. Make a conjecture about the meaning of your solutions to xercises 1 3 on page Identify at least two external resources that you could use to help you solve xercise 0 on page 8. ynamic eaching ools ynamic ssessment & rogress Monitoring ool Interactive Whiteboard Lesson Library ynamic lassroom with ynamic Investigations NW 1. Given square with diagonal, prove ; In this problem, the square could represent the baseball diamond, and then diagonal would represent the distance from home plate to second base. o, you could use this problem to prove the equivalent of H H. hen you could just redraw square with diagonal this time, so that is the equivalent of. It could easily be shown that the third triangle is congruent to the first two.. he theorems on page 75 correspond to other triangle congruence theorems given. H corresponds to, LL corresponds to, and L corresponds to or. 3. ample answer: a distance website and a map website ongruent triangles are often used to create company logos. Why are they used and what are the properties that make them attractive? ollowing the required constraints, create your new logo and justify how your shape contains the required properties. o explore the answers to these questions and more, go to igideasmath.com. erformance ask reating the Logo 89 hapter 5 89

81 NW 1. acute isosceles , , corresponding sides: GH LM, HJ MN, JK N, and GK L ; corresponding angles: G L, H M, J N, and K ; ample answer: JHGK NML hapter eview 5.1 ngles of riangles (pp ) lassify the triangle by its sides and by measuring its angles. he triangle does not have any congruent sides, so it is scalene. he measure of is 117, so the triangle is obtuse. he triangle is an obtuse scalene triangle. 1. lassify the triangle at the right by its sides and by measuring its angles. ind the measure of the exterior angle x ynamic olutions available at igideasmath.com x (9x + 9) ind the measure of each acute angle. 4. 8x 7x 5. (7x + 6) (6x 7) 5. ongruent olygons (pp ) Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. he diagram indicates that. orresponding angles,, orresponding sides,, 6. In the diagram, GHJK LMN. Identify all pairs of congruent corresponding parts. hen write another congruence statement for the quadrilaterals. K G H L M N 7. ind m V. U 74 J V 90 hapter 5 ongruent riangles 90 hapter 5

82 5.3 roving riangle ongruence by (pp ) Write a proof. Given, rove MN 1.. ON 1. Given. Given Vertical ngles ongruence heorem (heorem.6) ongruence heorem (heorem 5.5) NW 8. no; here are two pairs of congruent sides and one pair of congruent angles, but the angles are not the included angles. 9. ee dditional nswers. 10. ; 11. ; V 1. ; 13. ; V 14. x = 15, y = 5 ecide whether enough information is given to prove that WXZ YZX using the ongruence heorem (heorem 5.5). If so, write a proof. If not, explain why. 8. W X 9. W X Z Y Z Y 5.4 quilateral and Isosceles riangles (pp ) In LMN, LM LN. Name two congruent angles. LM LN, so by the ase ngles heorem (heorem 5.6), M N. L opy and complete the statement. 10. If, then. M N 11. If V V, then. 1. If, then. 13. If V V, then. V 14. ind the values of x and y in the diagram. 6 8x 5y + 1 hapter 5 hapter eview 91 hapter 5 91

83 NW 15. no; here is only enough information to conclude that two pairs of sides are congruent. 16. ee dditional nswers. 5.5 roving riangle ongruence by (pp ) Write a proof. Given, rove MN 1. ON 1. Given.. Given eflexive roperty of ongruence (heorem.1) ongruence heorem (heorem 5.8) 15. ecide whether enough information is given to prove that LM NM using the ongruence heorem (hm. 5.8). If so, write a proof. If not, explain why. M L N 16. ecide whether enough information is given to prove that WXZ YZX using the HL ongruence heorem (hm. 5.9). If so, write a proof. If not, explain why. W X Z Y 5.6 roving riangle ongruence by and (pp ) Write a proof. Given, rove MN ON Given.. Given Vertical ngles ongruence heorem (hm..6) ongruence heorem (hm. 5.11) 9 hapter 5 ongruent riangles 9 hapter 5

84 ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.11). If so, write a proof. If not, explain why. 17. G, HJK 18. UV, K G J H V U ecide whether enough information is given to prove that the triangles are congruent using the ongruence heorem (hm. 5.10). If so, write a proof. If not, explain why. 19. LN, LMN 0. WXZ, YZX 5.7 L N M Using ongruent riangles (pp. 77 8) xplain how you can prove that. If you can show that, then you will know that. You are given and. You know that by the eflexive roperty of ongruence (hm..1). wo pairs of sides and their included angles are congruent, so by the ongruence heorem (hm. 5.5),. W Z X Y NW 17. ee dditional nswers. 18. no; here is only enough information to conclude that one pair of angles and one pair of sides are congruent. 19. ee dditional nswers. 0. no; here is only enough information to conclude that one pair of angles and one pair of sides are congruent. 1. y the ongruence heorem (hm. 5.5), HJK LMN. ecause corresponding parts of congruent triangles are congruent, K N.. irst, state that V V. hen, use the ongruence heorem (hm. 5.8) to prove that V V. ecause corresponding parts of congruent triangles are congruent, V V. V 1 and V by the Vertical ngles ongruence heorem (hm..6). o, by the ransitive roperty of ongruence (hm..), 1. ecause corresponding parts of congruent triangles are congruent,. 1. xplain how to prove that K N. K M H L J N. Write a plan to prove that 1. 1 V U W hapter 5 hapter eview 93 hapter 5 93

85 NW 3. Using the istance ormula, O = h + k, = h + k, O = j, and = j. o, O and O. lso, by the eflexive roperty of ongruence (hm..1), O O. o, you can apply the ongruence heorem (hm. 5.8) to conclude that O O. 5.8 oordinate roofs (pp ) Write a coordinate proof. Given oordinates of vertices of O and rove O y (j, j) (j, j) 4. y ( p, 0) 5. (k, k) (0, k) (p, 0) x egments O and have the same length. O = ( j 0) + ( j 0) = j + j = j = j = ( j j) + ( j 0) = ( j) + j = j = j egments and have the same length. = = j = (j j) + (j j) = j + j = j = j egments O and have the same length. O(0, 0) (j, 0) x O = j 0 = j = j 0 = j o, you can apply the ongruence heorem (heorem 5.8) to conclude that O. 3. Write a coordinate proof. Given oordinates of vertices of quadrilateral O rove O O y (h, k + j) (0, j) (h, k) O(0, 0) x 4. lace an isosceles triangle in a coordinate plane in a way that is convenient for finding side lengths. ssign coordinates to each vertex. 5. rectangle has vertices (0, 0), (k, 0), and (0, k). ind the fourth vertex. 94 hapter 5 ongruent riangles 94 hapter 5

86 5 hapter est Write a proof. 1. Given. Given JK ML, MJ KL 3. Given, rove rove MJK KLM rove 4. ind the measure of each acute angle in the figure at the right. 5. Is it possible to draw an equilateral triangle that is not equiangular? If so, provide an example. If not, explain why. 6. an you use the hird ngles heorem (heorem 5.4) to prove that two triangles are congruent? xplain your reasoning. Write a plan to prove that Is there more than one theorem that could be used to prove that? If so, list all possible theorems. 1 y x M J L K Write a coordinate proof to show that the triangles created by the keyboard stand are congruent. 11. he picture shows the yramid of estius, which is located in ome, Italy. he measure of the base for the triangle shown is 100 oman feet. he measures of the other two sides of the triangle are both 144 oman feet. a. lassify the triangle shown by its sides. b. he measure of 3 is 40. What are the measures of 1 and? xplain your reasoning. If students need help... Lesson utorials kills eview Handbook igideasmath.com V W X (4x ) (3x + 8) Z Y N hapter 5 hapter est 95 If students got it... esources by hapter nrichment and xtension umulative eview erformance ask tart the next ection 1 3 NW 1 3. ee dditional nswers , no; y the orollary to the ase ngles heorem (or. 5.), if a triangle is equilateral, then it is also equiangular. 6. no; he hird ngles heorem (hm. 5.4) can be used to prove that two triangles are equiangular, but is not sufficient to prove that the triangles are congruent. You need to know that at least one pair of corresponding sides are congruent. 7. irst, use the HL ongruence heorem (hm. 5.9) to prove that. ecause corresponding parts of congruent triangles are congruent,. hen, use the ase ngles heorem (hm. 5.6) to prove that Use the ongruence heorem (hm. 5.8) to prove that VX ZX. Use the Vertical ngles ongruence heorem (hm..6) and the ongruence heorem (hm. 5.5) to prove that VXW ZXY. ecause corresponding parts of congruent triangles are congruent, ZX VX and W Y. Use the egment ddition ostulate (ost. 1.) to show that VY ZW. hen, use the ongruence heorem (hm. 5.10) to prove that VY ZW. ecause corresponding parts of congruent triangles are congruent, yes; HL ongruence heorem (hm. 5.9), ongruence heorem (hm. 5.10), ongruence heorem (hm. 5.11), ongruence heorem (hm. 5.5) 10. Using the istance ormula, = 18 and = 18. o,. lso, the horizontal segments and each have a slope of 0, which implies that they are parallel. o, intersects and to form congruent alternate interior angles, and. y the Vertical ngles ongruence heorem (hm..6),. o, by the ongruence heorem (hm. 5.11),. 11. ee dditional nswers. hapter 5 95