Triangles and Congruence

Transcription

1 HPT 5 Triangles and ongruence What You ll Learn Key Ideas Identify parts of triangles and classify triangles by their parts. (Lesson 5 1) Use the ngle Sum Theorem. (Lesson 5 2) Identify translations, reflections, and rotations and their corresponding parts. (Lesson 5 3) Name and label corresponding parts of congruent triangles. (Lesson 5 4) Use the SSS, SS, S, and S tests for congruence. (Lessons 5 5 and 5 6) Key Vocabulary congruent triangles (p. 203) right triangle (p. 188) triangle (p. 188) Why It s Important Social Studies The Hopi people have lived on three isolated mesas in what is now northern rizona for more than a thousand years. Oraibi, a Hopi village on Third Mesa, is the oldest continuously inhabited village in the United States. The Hopi live in multilevel adobe or stone villages called pueblos. Triangles are the simplest of the polygons. You will determine how triangles are used to create the design on Hopi pottery in Lesson hapter 5 Triangles and ongruence

2 Study these lessons to improve your skills. heck Your eadiness Use a protractor to draw an angle having each measurement. Then classify each angle as acute, obtuse, or right. Lesson 3 2, pp Lesson 3 5, pp etermine the measures of the complement and supplement of each angle Solve each equation. heck your solution. lgebra eview, p. 722 lgebra eview, p n x m g k q b g y c (n 4) n x 4x 6x 180 Make this oldable to help you organize your hapter 5 notes. egin with 3 sheets of 1 plain 8 2 " by 11" paper. ➊ old in half lengthwise. ➋ old the top to the bottom. ➌ Open and cut along ➍ Label each tab as shown. the second fold to make two tabs. Triangles classified by ngles Triangles classified by Sides eading and Writing s you read and study the chapter, write what you learn about the two methods of classifying triangles under the tabs. hapter 5 Triangles and ongruence 187

3 5 1 What You ll Learn You ll learn to identify the parts of triangles and to classify triangles by their parts. Why It s Important rt bstract artists use geometric shapes in their designs. See xercise 24. lassifying Triangles Optical art is a form of abstract art that creates special effects by using geometric patterns. The design at the right looks like a spiral staircase, but it is made mostly of triangles. In geometry, a triangle is a figure formed when three noncollinear points are connected by segments. ach pair of segments forms an angle of the triangle. The vertex of each angle is a vertex of the triangle. Triangles are named by the letters at their vertices. Triangle, written, is shown below. vertex angle ead the symbol as triangle. Other names for are,,,, and. The sides are,, and. The vertices are,, and. The angles are,, and. side In hapter 3, you classified angles as acute, obtuse, or right. Triangles can also be classified by their angles. ll triangles have at least two acute angles. The third angle is either acute, obtuse, or right. acute Triangles lassified by ngles right all acute angles 188 hapter 5 Triangles and ongruence obtuse 43 one obtuse angle one right angle

4 Triangles can also be classified by their sides. n equal number of slashes on the sides of a triangle indicate that those sides are congruent. scalene isosceles equilateral no sides congruent at least two sides congruent all sides congruent Triangles lassified by Sides Since all sides of an equilateral triangle are congruent, then at least two of its sides are congruent. So, all equilateral triangles are also isosceles triangles. Some parts of isosceles triangles have special names. The angle formed by the congruent sides is called the vertex angle. The congruent sides leg are called legs. The two angles formed by the base and one of the congruent sides are called base angles. leg base angle base angle The side opposite the vertex angle is called the base. xamples lassify each triangle by its angles and by its sides G G is a right isosceles triangle. is an acute equilateral triangle. Your Turn a. X b. 25 L Z Y M N Lesson 5 1 lassifying Triangles 189

5 xample 3 lgebra Link ind the measures of and of isosceles triangle if is the vertex angle. xplore You know that is the vertex angle. Therefore,. 5x 7 Since,. You can write and solve an equation. Plan lgebra eview Solving Multi-Step quations, p x 5 efinition of congruent segments 5x 7 23 Substitution 5x dd 7 to each side. 5x 30 Simplify. Solve 5x ivide each side by 5. x6 Simplify. To find the measures of and, replace x with 6 in the expression for each measure. 5x 7 5(6) or 23 3x 5 3(6) or 13 Therefore, 23 and 13. xamine Since 23 and 23, the triangle is isosceles. heck for Understanding ommunicating Mathematics 1. raw a scalene triangle. 2. Sketch and label an isosceles triangle in which the vertex angle is X and the base is YZ. 3. Is an equilateral triangle also an isosceles triangle? xplain why or why not. Guided Practice lassify each triangle by its angles and by its sides. xamples 1 & cm triangle vertex equilateral isosceles scalene 6. 8 ft 45 8 ft 5.5 cm cm xample 3 8 ft is an isosceles triangle 7. lgebra with base. ind and x 1 2x hapter 5 Triangles and ongruence

6 xercises Practice 4.8 in See xamples 1, 2 1, xtra Practice See page m cm 11 m 11 m 7.7 m in m 7 in. Homework Help lassify each triangle by its angles and by its sides. 8. or xercises 7 in in. 12 in. 10 m cm cm Triangle XYZ has angles that measure 30,, and 90. lassify the triangle by its angles. Make a sketch of each triangle. If it is not possible to sketch the figure, write not possible acute isosceles right equilateral obtuse and not isosceles right and not scalene obtuse equilateral pplications and Problem Solving 23. rchitecture efer to the photo at the right. lassify each triangle by its angles and by its sides. a. b. c. 24. rt efer to the ata Update or optical art design on the latest information about page 188. lassify the optical art, visit: triangles by their angles and by their sides. lcoa Office uilding, San rancisco, Lesson 5 1 lassifying Triangles 191

7 25. Quilting lassify the triangles that are used in the quilt blocks. a. b. Ohio Star uck s oot in the Mud 26. lgebra is an equilateral triangle in which x 5, 3x 3, and 2x 1. a. raw and label. b. ind the measure of each side. 27. lgebra ind the measure of each side of isosceles triangle if is the vertex angle and the perimeter of the triangle is 20 meters. 2x x 2x 28. ritical Thinking Numbers that can be represented by a triangular arrangement of dots are called triangular numbers. The first four triangular numbers are 1, 3, 6, and 10. ind the next two triangular numbers. Write an equation in slope-intercept form of the line with the given slope that passes through the given point. (Lesson 4 6) Mixed eview 29. m 3, (0, 4) 30. m 0, (0, 2) 31. m 2, ( 2, 1) ind the slope of the lines passing through each pair of points. (Lesson 4 5) 32. (5, 7), (4, 5) 35. Sports In the Olympic ski-jumping competition, the skier tries to make the angle between his body and the front of his skis as small as possible. If a skier is aligned so that the front of his skis makes a 20 angle with his body, what angle is formed by the tail of the skis and his body? (Lesson 3 5) 20 Standardized Test Practice 34. (5, 2), (5, 1) 33. (8, 4), ( 2, 4) 36. Multiple hoice Use the number line to find. (Lesson 2 1) hapter 5 Triangles and ongruence

8 5 2 What You ll Learn You ll learn to use the ngle Sum Theorem. Why It s Important ngles of a Triangle If you measure and add the angles in any triangle, you will find that the sum of the angles have a special relationship. ut and fold a triangle as shown below. Make a conjecture about the sum of the angle measures of a triangle. onstruction uilders use the measure of the vertex angle of an isosceles triangle to frame buildings. See xercise 21. P Q 2 Q Q P P You can use a graphing calculator to verify your conjecture. Step 1 Use the Triangle tool on the 2 menu. Move the pencil cursor to each location where you want a vertex and press NT. The calculator automatically draws the sides. Label the vertices,, and. Step 2 Use the ngle tool under Measure on the 5 menu to measure each angle. Graphing alculator Tutorial See pp Try These 1. etermine the sum of the measures of the angles of your triangle. 2. rag any vertex to a different location, measure each angle, and find the sum of the measures. 3. epeat xercise 2 several times. 4. Make a conjecture about the sum of the angle measures of any triangle. The results of the activities above can be stated in the ngle Sum Theorem. Words: The sum of the measures of the angles of a triangle is 180. Theorem 5 1 ngle Sum Model: Theorem Symbols: x y x y z 180 z Lesson 5 2 ngles of a Triangle 193

9 You can use the ngle Sum Theorem to find missing measures in triangles. xamples 1 ind mt in ST. m ms mt mt mt mt mt 59 lgebra eview Solving One-Step quations, p ngle Sum Theorem 54 Substitution dd. Subtract from each side. S T ind the value of each variable in. y and are vertical angles. Vertical angles are congruent, so m m. Therefore, x 85. Now find the value of y. m m m y y y y x 55 ngle Sum Theorem Substitution dd. Subtract 140 from each side. Simplify. Therefore, x 85 and y 40. Your Turn a a. ind ml in MNL if mm 25 and mn 25. b. ind the value of each variable in the figure at the right. You can use the ngle Sum Theorem to discover a relationship between the acute angles of a right triangle. In ST, is a right angle. 28 b 65 S T omplementary ngles: Lesson 3 5 m mt ms mt ms mt ms mt ms 90 ngle Sum Theorem Substitution Subtract 90 from each side. Simplify. y the definition of complementary angles, T and S are complementary. This relationship is stated in the following theorem. 194 hapter 5 Triangles and ongruence

10 Words: The acute angles of a right triangle are complementary. Theorem 5 2 Model: Symbols: x y = 90 x y xample lgebra Link 3 ind m and m in right triangle. m m 90 2x 3x 90 5x 90 5x 5 2x Theorem 5 2 Substitution ombine like terms ivide each side by 5. x 18 Simplify. 3x Now replace x with 18 in the expression for each angle. m 2x m 3x 2(18) or 36 3(18) or 54 n equiangular triangle is a triangle in which all three angles are congruent. You can use the ngle Sum Theorem to find the measure of each angle in an equiangular triangle. Triangle PQ is an equiangular triangle. Since mp mq m, the measure of each angle of PQ is or 60. P Q This relationship is stated in Theorem 5 3. Words: The measure of each angle of an equiangular triangle is 60. Theorem 5 3 Model: Symbols: x = 60 x x x Lesson 5 2 ngles of a Triangle 195

11 heck for Understanding ommunicating Mathematics 1. hoose the numbers that are not measures of the three angles of a triangle. a. 10, 20, 150 b. 30, 60, 90 c. 40, 70, 80 d. 45, 55, 80 equiangular triangle 2. xplain how to find the measure of the third angle of a triangle if you know the measures of the other two angles. 3. Guided Practice xamples 1 & 2 Is it possible to have two obtuse angles in a triangle? Write a few sentences explaining why or why not. ind the value of each variable y y x b ind the value of each variable x Homework Help or xercises See xamples 8 12, 20, , lgebra The measures of the angles of a triangle are 2x, 3x, and 4x. ind the measure of each angle. xercises Practice 63 a 58 xample x 45 x x 50 3 xtra Practice x See page 734. a x 14. y x 80 x b x x z 51 y z hapter 5 Triangles and ongruence y

12 ind the measure of each angle in each triangle (x 20) 19. 2x x 50 (x 15) x 20. The measure of one acute angle of a right triangle is 25. ind the measure of the other acute angle. pplications and Problem Solving 21. onstruction The roof lines of many buildings are shaped like the legs of an isosceles triangle. ind the measure of the vertex angle of the isosceles triangle shown at the right lgebra The measures of the angles of a triangle are x + 5, 3x 14, and x ind the measure of each angle ritical Thinking If two angles of one triangle are congruent to two angles of another triangle, what is the relationship between the third angles of the triangles? xplain your reasoning. Mixed eview 24. The perimeter of GHI is 21 units. ind GH and GI. (Lesson 5 1) G x3 25. State the slope of the lines perpendicular to the graph of y 3x 2. (Lesson 4 6) H x4 I 8 xercise 24 Identify each pair of angles as alternate interior, alternate exterior, consecutive interior, or vertical. (Lesson 4 2) 26. 1, , , , 15 Standardized Test Practice 30. Short esponse Points X, Y, and Z are collinear, and XY 45, YZ 23, and XZ 22. Locate the points on a number line. (Lesson 2 2) Lesson 5 2 ngles of a Triangle 197

13 5 3 What You ll Learn You ll learn to identify translations, reflections, and rotations and their corresponding parts. Geometry in Motion We live in a world of motion. Geometry helps us define and describe that motion. In geometry, there are three fundamental types of motion: translation, reflection, and rotation. Translation eflection otation Why It s Important rt rtists use motion geometry to make designs. See xample 6. line fixed point In a translation, you slide a figure from one position to another without turning it. Translations are sometimes called slides. In a reflection, you flip a figure over a line. The new figure is a mirror image. eflections are sometimes called flips. In a rotation, you turn the figure around a fixed point. otations are sometimes called turns. When a figure is translated, reflected, or rotated, the lengths of the sides of the figure do not change. xamples Identify each motion as a translation, reflection, or rotation. 2 1 rotation 3 reflection translation Your Turn a. 198 hapter 5 Triangles and ongruence b. c.

14 The figure below shows a translation. ach point on the original figure is called a preimage. ead as triangle maps to triangle. Its matching point on the corresponding figure is called its image. ach point on the preimage can be paired with exactly one point on its image, and each point on the image can be paired with exactly one point on the preimage. This one-to-one correspondence is an example of a mapping. The symbol is used to indicate a mapping. In the figure,. In naming the triangles, the order of the vertices indicates the corresponding points. Preimage Image Preimage Image This mapping is called a transformation. In the figure, XYZ by a reflection. xamples 4 Name the image of X. X XYZ X corresponds to. 5 So, is the image of X. Z Y Name the side that corresponds to. Point corresponds to point X. XYZ Point corresponds to point Y. XY So, corresponds to. Your Turn In the figure, LMN QS by a rotation. M d. Name the image of M. e. Name the angle that corresponds to S. f. Name the image of LM. g. Name the side that corresponds to L N. L N Q Lesson 5 3 Geometry in Motion S 199

15 eal World xample rt Link Translations, reflections, and rotations are all isometries. n isometry is a movement that does not change the size or shape of the figure being moved. rtists often use isometries in designs. One of the most famous artists to use this technique was M.. scher. 6 Identify the type of transformation in the artwork at the right. ach figure can be moved to match another without turning or flipping. Therefore, the motion is a translation. M.. scher, Pegasus heck for Understanding ommunicating Mathematics Guided Practice xamples xplain the difference between a translation and a rotation. 2. Suppose ST. ntonio says that corresponds to T. Keisha says she needs to see the drawing to know which angles correspond. Who is correct? xplain your reasoning. Identify each motion as a translation, reflection, or rotation translation reflection rotation transformation preimage isometry image mapping xamples 4 & 5 In the figure at the right, XYZ ST. X 6. Name the image of X Y. 7. Name the angle that corresponds to. Z Y T S xample 6 8. Native merican esigns The design below was found on food bowls that were discovered in the ruins of an ancient Hopi pueblo. Identify the transformations in the design. 200 hapter 5 Triangles and ongruence

16 xercises Practice Identify each motion as a translation, reflection, or rotation Homework Help or xercises See xamples 1 3 4, 5 6 xtra Practice See page 734. In the figure at the right, MNP GH. M 18. Which angle corresponds to N? 19. Which side corresponds to MN? N 20. Name the angle that corresponds to H. 21. Name the image of point Q. 22. Name the side that corresponds to GH. 23. Name the image of PQ. 25. ngines ams are important parts of engines because they change motion from one direction to another. s the cam turns around, the pistons move up and down. Identify the transformation that occurs in the cams. J P H 24. If PQ, which angle corresponds to? pplications and Problem Solving Q K G piston cam Lesson 5 3 Geometry in Motion 201

17 26. rt The figure at the left shows an untitled work by M.. scher. Identify the type of transformation used to complete the work. 27. ritical Thinking The transformation below is called a glide reflection. How is this transformation different from a translation, reflection, and rotation? M.. scher, lying ish Mixed eview 28. The measure of one acute angle of a right triangle is 30. ind the measure of the other acute angle. (Lesson 5 2) 29. lgebra XYZ is an equilateral triangle in which XY 2x 2, YZ x 7, and XZ 4x 8. ind the measure of each side. (Lesson 5 1) raw a figure for each pair of planes or segments. (Lesson 4 1) 30. parallel planes 31. skew segments 32. intersecting planes Standardized Test Practice 33. Multiple hoice Which ordered pair represents the intersection of line t and line m? (Lesson 2 4) (2, 3) ( 2, 3) (2, 3) ( 2, 3) y t m O x > Quiz 1 Lessons 5 1 through 5 3 lassify each triangle by its angles and by its sides. (Lesson 5 1) in in. 4. lgebra The measures of the angles of a triangle are 2x, 5x, and 5x. ind the measure of each angle. (Lesson 5 2) 5. Identify the motion as a translation, reflection, or rotation. (Lesson 5 3) 20 9 in hapter 5 Triangles and ongruence

18 5 4 What You ll Learn You ll learn to identify corresponding parts of congruent triangles. ongruent Triangles You ve learned that congruent segments have the same length and congruent angles have the same degree measure. In the following activity, you will learn about congruent triangles. Why It s Important rafts The pieces of Materials: scissors On a piece of grid paper, draw two triangles like the ones below. Label the vertices as shown. ut out the triangles. Put one triangle over the other so that the parts with the same measures match up. Step 2 straightedge Step 1 grid paper fabric used to make a quilt are congruent to a template. See xercise 27. Try These 1. Identify all of the pairs of angles and sides that match or correspond. 2. Triangle is congruent to. What is true about their corresponding sides and angles? rcs are used to show which angles are congruent. Slash marks are used to show which sides are congruent. If a triangle can be translated, rotated, or reflected onto another triangle so that all of the vertices correspond, the triangles are congruent triangles. The parts of congruent triangles that match are called corresponding parts. In the figure,. s in a mapping, the order of the vertices indicates the corresponding parts. ongruent ngles ongruent Sides These relationships help to define congruent triangles. Lesson 5 4 ongruent Triangles 203

19 efinition of ongruent Triangles (PT) If the corresponding parts of two triangles are congruent, then the two triangles are congruent. If two triangles are congruent, then the corresponding parts of the two triangles are congruent. PT is an abbreviation for orresponding Parts of ongruent Triangles are ongruent. xamples 1 If PQ MLN, name the congruent angles and sides. Then draw the triangles, using arcs and slash marks to show the congruent angles and sides. irst, name the three pairs of congruent angles by looking at the order of the vertices in the statement PQ MLN. P M Q L N So, P M, Q L, and N. ML Since P corresponds to M, and Q corresponds to L, PQ. LN Since Q corresponds to L, and corresponds to N, Q. MN Since P corresponds to M, and corresponds to N, P. P 2 M Q L N The corresponding parts of two congruent triangles are marked on the figure. Write a congruence statement for the two triangles. List the congruent angles and sides. I K G J GHI JHK H KH I GH JH GI JK J I H K G The congruence statement can be written by matching the vertices of the congruent angles. Therefore, IGH KJH. Your Turn The corresponding parts of two congruent triangles are marked on the figure. Write a congruence statement for the two triangles. 204 hapter 5 Triangles and ongruence

20 xample ST is congruent to XYZ. ind the value of n. 3 lgebra Link Z S (2n 10) 90 Y T X Since ST XYZ, the corresponding parts are congruent. m S m Y 50 2n 10 Substitution n Subtract 10 from each side. 40 2n Simplify. lgebra eview Solving Multi-Step quations, p n 2 ivide each side by n Simplify. heck for Understanding ommunicating Mathematics 1. xplain what it means when one triangle is congruent to another. 2. escribe how transformations are used to determine whether triangles are congruent. Getting eady Guided Practice If, name the corresponding side or angle. Sample: Solution: corresponds to. 3. xample 1 congruent triangles corresponding parts If XYZ, name the congruent angles and sides. Then draw the triangles, using arcs and slash marks to show the congruent angles and sides. 6. Y xample 2 Z X omplete each congruence statement. 8.? 9.? Lesson 5 4 ongruent Triangles 205

21 xample lgebra QP is congruent to ONM. ind the value of x. O N (3x 5) m 70 m P xercises Homework Help 11, 12, 19 23, 26, 27 M or each pair of congruent triangles, name the congruent angles and sides. Then draw the triangles, using arcs and slash marks to show the congruent angles and sides. Practice or xercises Q T See xamples , 25 3 Q S V xtra Practice See page 735. U QS TUV omplete each congruence statement ?? ? 17. T? 18. S TS? 206 hapter 5 Triangles and ongruence?

22 If GH, name the part that is congruent to each angle or segment G 22. G 24. If PQ YXZ, m P 63, and m Q 57, find m X. pplications and Problem Solving 25. lgebra If HG, what is the value of x? 6 H 5x 10 G Landscaping Two triangular gardens have the same size and shape. The landscaper needed 24 feet of fencing for one garden. How much fencing is needed for the second garden? xplain your reasoning. 27. rafts Many quilts are designed using triangles. Quilters start with a template and trace around the template, outlining the triangles to be cut out. xplain why the triangles are congruent. template fabric triangles 28. ritical Thinking etermine whether each statement is true or false. If true, explain your reasoning. If false, show a counterexample. a. If two triangles are congruent, their perimeters are equal. b. If two triangles have the same perimeter, they are congruent. Mixed eview Identify each motion as a translation, reflection, or rotation. (Lesson 5 3) ommunication support cable called a guy wire is attached to a utility pole to give it stability. Safety regulations require a minimum angle of 30 between the pole and the guy wire. etermine the measure of the angle between the guy wire and the ground. (Lesson 5 2) Standardized Test Practice guy wire Short esponse If m 45, classify as acute, right, or obtuse. (Lesson 3 2) 34. Multiple hoice hoose the false statement. (Lesson 1 3) Two points determine two lines. line contains at least two points. Three points that are not on the same line determine a plane. If two planes intersect, then their intersection is a line. Lesson 5 4 ongruent Triangles 207

23 hapter 5 Materials Investigation Introducing the ongruence Postulates patty paper scissors Is it possible to show that two triangles are congruent without showing that all six pairs of corresponding parts are congruent? Let s look for a shortcut. straightedge Investigate 1. Use patty paper to investigate three pairs of congruent sides. a. raw a triangle on a piece of patty paper. b. opy the sides of the triangle onto another piece of patty paper and cut them out. c. rrange the pieces so that they form a triangle. d. Is this triangle congruent to the original triangle? xplain your reasoning. e. Try to form another triangle. Is it congruent to the original triangle? f. an three pairs of congruent sides be used to show that two triangles are congruent? 208 hapter 5 Triangles and ongruence

24 2. Use patty paper to investigate three pairs of congruent angles. a. raw a triangle on a piece of patty paper. b. opy each angle of the triangle onto a separate piece of patty paper and cut them out. xtend each ray of each angle to the edge of the patty paper. c. rrange the pieces so that they form a triangle. d. Is this triangle congruent to the original triangle? xplain your reasoning. e. Try to form another triangle. Is this triangle congruent to the original triangle? f. an three pairs of congruent angles be used to show that two triangles are congruent? In this investigation, you will determine which three pairs of corresponding parts can be used to show that two triangles are congruent. Use patty paper or graphing software to investigate these six cases. (You have already investigated the first two.) 1. three pairs of congruent sides 2. three pairs of congruent angles 3. two pairs of congruent sides and the pair of congruent angles between them 4. two pairs of congruent sides and one pair of congruent angles not between them 5. two pairs of congruent angles and the pair of congruent sides between them 6. two pairs of congruent angles and one pair of congruent sides not between them Presenting Your onclusions Here are some ideas to help you present your conclusions to the class. Make a poster that summarizes your results. Make a model with straws that illustrates why certain pairs of corresponding parts cannot be used to show that two triangles are congruent. e sure to show counterexamples. Investigation or more information on the congruence postulates, visit: hapter 5 Investigation Take a Shortcut 209

25 5 5 What You ll Learn You ll learn to use the SSS and SS tests for congruence. SSS and SS Triangles are common in construction, because triangles, unlike squares, maintain thair shape under stress. You can see this yourself if you use straws and a string to make a triangle and a four-sided figure. Why It s Important onstruction rchitects add strength to their buildings by using triangles for support. See xercise 7. rigid not rigid This rigidity hints at an underlying geometric concept: a triangle with three sides of a set length has exactly one shape. onstruction Materials: compass straightedge Step 1 raw an acute scalene triangle on a piece of paper. Label its vertices,, and on the interior of each angle. Step 2 onstruct a segment congruent to. Label the endpoints of the segment and. Step 3 djust the compass setting to the length of. Place the compass at point and draw a large arc above. Step 4 djust the compass setting to the length of. Place the compass at point and draw an arc to intersect the one drawn from point. Label the intersection. Step 5 raw and. scissors Step 1 Step 2 Step 4 Try These 1. Label the vertices of on the interior of each angle. Then cut out the two triangles. Make a conjecture. re the triangles congruent? 2. If the triangles are congruent, write a congruence statement. 3. Verify your conjecture with another triangle. 210 hapter 5 Triangles and ongruence

26 In the previous activity, you constructed a congruent triangle by using only the measures of its sides. This activity suggests the following postulate. Words: If three sides of one triangle are congruent to three corresponding sides of another triangle, then the triangles are congruent. Model: S Postulate 5 1 SSS Postulate The abbreviation SSS is read as Side-Side-Side. T Symbols: If S, ST, and T, then ST. xample 1 ML MN Q NL In two triangles, P Q, P, and. Write a congruence statement for the two triangles. raw a pair of congruent triangles. Identify the congruent parts with slashes. Label the vertices of one triangle. P Use the given information to label the vertices of the second triangle. y SSS, PQ MLN. Q L Q P M N Your Turn a. In two triangles, ZY, X Y, and X Z. Write a congruence statement for the two triangles. In a triangle, the angle formed by two given sides is called the included angle of the sides. is the included angle of and. is the included angle of and. is the included angle of and. Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent. You can also show their congruence by using two sides and the included angle. Lesson 5 5 SSS and SS 211

27 Words: Postulate 5 2 SS Postulate The abbreviation SS is read as Side-ngleSide. If two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the triangles are congruent. Model: S T Symbols: If S,, and T, then ST. xample 2 etermine whether the triangles shown at the right are congruent. If so, write a congruence statement and explain why the triangles are congruent. If not, explain why not. M X Y There are two pairs of congruent sides, YZ XZ N O and M O. There is one N pair of congruent angles, O Z, which is included between the sides. Therefore, MNO XYZ by SS. O Your Turn Q Z b. etermine whether the triangles shown at the right are congruent by SS. If so, write a congruence statement and tell why the triangles are congruent. If not, explain why not. P heck for Understanding ommunicating Mathematics 1. Sketch and label a triangle in which X is the included angle of YX and Z X. 2. Guided Practice xample 1 included angle Karen says that there is only one triangle with sides of 3 inches, 4 inches, and 5 inches. Mika says that there can be many different triangles with those measures. Who is correct? xplain your reasoning. Write a congruence statement for each pair of triangles represented. UW UV WV 3. T, S, T S 212 hapter 5 Triangles and ongruence 4. GH HI, H,

28 xample 2 etermine whether each pair of triangles is congruent. If so, write a congruence statement and explain why the triangles are congruent onstruction Most roofs on residential buildings are made of triangular roof trusses. xplain how the SSS postulate guarantees that the triangles in the roof truss will remain rigid. xample 1 Gusset Upper chord 12 5 Lower chord Web member xercises Homework Help 8 11, , 16 19, 22 Write a congruence statement for each pair of triangles represented. Practice or xercises See xamples JK MN ON, L K, K N 10. X Y, X Z, X 9.,, 11. G H T S TS, G I, H I etermine whether each pair of triangles is congruent. If so, write a congruence statement and explain why the triangles are congruent xtra Practice See page Use the given information to determine whether the two triangles are congruent by SS. Write yes or no ,,,,,,,, Lesson 5 5 SSS and SS 213

29 pplications and Problem Solving 20. arpentry Suppose you are building a rectangular bookcase. How could you provide additional support so that the back of the bookcase won t shift? 21. Landscaping When small trees are planted, they are usually supported with a wooden stake as shown at the right. xplain how the stake provides support against the wind. 22. ritical Thinking Name the additional corresponding part needed to prove that the triangles below are congruent by SS. U xercise 21 S Mixed eview T W V 23. If PQ, m P 45, and m 38, find m. (Lesson 5 4) button 24. Word Processing The in some computer programs makes the indicated change in the position of the word Hello. Identify the change as a rotation, reflection, or translation. (Lesson 5 3) The coordinates of the endpoints of a segment are given. ind the coordinates of the midpoint of each segment. (Lesson 2 5) 25. ( 1, 2), ( 3, 8) Standardized Test Practice 27. (0, 0), (x, y) 28. Multiple hoice xpress in scientific notation. (lgebra eview) Quiz 2 > 26. (4, 8), ( 3, 4) Lessons 5 4 and esign Which triangles in the figure appear to be congruent? (Lesson 5 4) 2. If XYZ ST, which angle is congruent to S? (Lesson 5 4) 3. In two triangles, X Z, Y Z, and Y X. Write a congruence statement for the two triangles. (Lesson 5 5) xercise 1 etermine whether each pair of triangles is congruent. If so, write a congruence statement and explain why the triangles are congruent. (Lesson 5 5) M N P Q L 214 hapter 5 Triangles and ongruence

30 5 6 What You ll Learn You ll learn to use the S and S tests for congruence. S and S The side of a triangle that falls between two given angles is called the included side of the angles. It is the one side common to both angles. Why It s Important is the included side of and. is the included side of and. Surveying Surveyors use the S Postulate when setting up sight markers. See xercise 10. is the included side of and. You can show that two triangles are congruent by using two angles and the included side of the triangles. Words: Postulate 5 3 S Postulate The abbreviation S is read as ngle-sidengle. If two angles and the included side of one triangle are congruent to the corresponding angles and included side of another triangle, then the triangles are congruent. Model: S T Symbols: If, T, and T, then ST. xample 1 KL In PQ \ KJL, K, Q, and Q L. Write a congruence statement for the two triangles. egin by drawing a pair of congruent triangles. Mark the congruent parts with arcs and slashes. Label the vertices of one triangle P, Q, and. Locate K and L on the unlabeled triangle in the same positions as and Q. The unassigned vertex must be J. P Q K Therefore, PQ JLK by S. Your Turn J NL a. In and LMN, N,, and L. Write a congruence statement for the two triangles. Lesson 5 6 S and S L 215

31 The abbreviation S is read as ngle-ngleside. The ngle-ngle-side Theorem is called a theorem because it can be derived from the S Postulate. In S, the S is not between the two given angles. Therefore, the S indicates a side that is not included between the two angles. and are nonincluded sides of and. Words: Theorem 5 4 S Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent. Model: S T Symbols: If, T, and ST, then ST. xample 2 and each have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by S? If and are marked congruent, then and would be included sides. However, S requires the nonincluded sides. Therefore, and must be marked congruent. Your Turn b. and LMN each have one pair of sides and one pair of angles marked to show congruence. What other pair of angles must be marked so that the two triangles are congruent by S? c. What other pair of angles must be marked so that the two triangles are congruent by S? L M N 216 hapter 5 Triangles and ongruence

32 xamples etermine whether each pair of triangles is congruent by SSS, SS, S, or S. If it is not possible to prove that they are congruent, write not possible. 3 4 M P N O Q There are two pairs of congruent angles, and. There is one pair of corresponding congruent sides,, which is not included between the angles. There are two pairs of congruent P Q sides, MN and N O. There is one pair of congruent angles, M P, which is not included between the sides. Therefore, by S. Since SS is not a test for congruence, it is not possible to show the triangles are congruent from this information. Your Turn H d. e. L G I J K N M X L Z Y heck for Understanding ommunicating Mathematics 1. Sketch and label triangle XYZ in which X Z is an included side. Then name the two angles XZ is between. included side 2. xplain how you could construct a triangle congruent to a given triangle using S. 3. Guided Practice xample 1 Write a few sentences explaining the SSS, SS, S, and S tests for congruence. Give an example of each. Write a congruence statement for each pair of triangles represented. T 4. In and ST,, T, and. YZ 5. In and XYZ, X, Y, and. Lesson 5 6 S and S 217

33 xample 2 Name the additional congruent parts needed so that the triangles are congruent by the postulate or theorem indicated. 6. S 7. S xamples 3 & 4 etermine whether each pair of triangles is congruent by SSS, SS, S, or S. If it is not possible to prove that they are congruent, write not possible Surveying land , yd Write a congruence statement for each pair of triangles represented. Homework Help or xercises 10. Surveying Two surveyors 560 yards apart sight a marker on the other side of a canyon at angles of 27 and 38. What will happen if they repeat their measurements from the same positions on another day? xplain your reasoning. xample 1 xercises Practice See xamples 1 2 3, 4 xtra Practice In QS and TUV, Q T, S U, and QS TU. In and,,, and. XZ In ST and XYZ, S X, ST, and T Z. Q In MNO and PQ, M P, N, and NO. Name the additional congruent parts needed so that the triangles are congruent by the postulate or theorem indicated. 15. S 16. S See page S 18. S 218 hapter 5 Triangles and ongruence

34 etermine whether each pair of triangles is congruent by SSS, SS, S, or S. If it is not possible to prove that they are congruent, write not possible H J G S T X Y L T J pplications and Problem Solving Mixed eview U K Z V 23. Math History The figure shows how the Greek mathematician Thales ( ) determined the distance from the shore to enemy ships during a war. He sighted the ship from point P and then duplicated the angle at QPT. The angles at point Q are right angles. xplain why QT represents the distance from the shore to the ship. 24. ritical Thinking In ST and UVW, U, S V, and T UW. So, ST UVW by S. Prove ST UVW by S. Shore P Q (igure not drawn to scale.) T 25. In two triangles, MN PQ MO P,, and N O Q. Write a congruence statement for the two triangles and explain why the triangles are congruent. (Lesson 5 5) If HT MNP, complete each statement. (Lesson 5 4) Standardized Test Practice 29. Multiple hoice The graph shows the sales of athletic and sports equipment from 1995 to etween which two years was the percent of increase the greatest? (Statistics eview) 1996 to to to to thletic and Sports quipment Sales (billions) Year 26.? 27. H T? 28. P? Sales Source: National Sporting Goods ssociation Lesson 5 6 S and S 219

35 HPT 5 Study Guide and ssessment Understanding and Using the Vocabulary fter completing this chapter, you should be able to define each term, property, or phrase and give an example or two of each. acute triangle (p. 188) base (p. 189) base angles (p. 189) congruent triangle (p. 203) corresponding parts (p. 203) equiangular triangle (p. 195) equilateral triangle (p. 189) image (p. 199) included angle (p. 211) included side (p. 215) isometry (p. 200) isosceles triangle (p. 189) legs (p. 189) mapping (p. 199) obtuse triangle (p. 188) preimage (p. 199) reflection (p. 198) right triangle (p. 188) eview ctivities or more review activities, visit: rotation (p. 198) scalene triangle (p. 189) transformation (p. 199) translation (p. 198) triangle (p. 188) vertex (p. 188) vertex angle (p. 189) State whether each sentence is true or false. If false, replace the underlined word(s) to make a true statement Triangles can be classified by their angles and sides. n isosceles triangle has two vertex angles. The sum of the measures of the angles of a triangle is 360. n equiangular triangle is defined as a triangle with three congruent sides. The acute angles of a right triangle are supplementary. SSS, SS, S, and S are ways to show that two triangles are congruent. translation is an example of a transformation. n equilateral triangle is also an isosceles triangle. S refers to two angles and their included side. eflections are sometimes called turns. Skills and oncepts Objectives and xamples Lesson 5 1 Identify the parts of triangles and classify triangles by their parts. 7.5 m 7.5 m lassify each triangle by its angles and by its sides in eview xercises 6m 8 in. 8 in. 66 The triangle is acute and isosceles. 220 hapter 5 Triangles and ongruence

36 hapter 5 Study Guide and ssessment Objectives and xamples eview xercises Lesson 5 2 Use the ngle Sum Theorem. ind the value of each variable x y ind m in a m + m m = 180 m = 180 m = 180 m = m = b Lesson 5 3 Identify translations, reflections, and rotations and their corresponding parts T S Lesson 5 4 Name and label corresponding parts of congruent triangles. ST corresponds to. Write a congruence statement for the two triangles. 30 Suppose. ST by a translation. is the image of. y Name the angle that corresponds to. Name the image of. Name the image of. Identify the transformation that occurred in the mapping. omplete each congruence statement T N S O M V Q W P MNO? ST? hapter 5 Study Guide and ssessment 221

37 hapter 5 Study Guide and ssessment Mixed Problem Solving See pages Objectives and xamples eview xercises Lesson 5 5 Use the SSS and SS tests for congruence. S M L T etermine whether each pair of triangles is congruent. If so, write a congruence statement and explain why the triangles are congruent. 22. X Z 23. Y N ST LNM by SS. P Lesson 5 6 Use the S and S tests for congruence. X G Q etermine whether each pair of triangles is congruent by SSS, SS, S, or S. If it is not possible to prove that they are congruent, write not possible L M P M Z Y H K L T O N XYZ GH by S. S pplications and Problem Solving 26. Maps lassify the triangle by its sides. (Lesson 5 1) Tallahassee 27. lgebra ind the measure of in. (Lesson 5 2) (x 32) 58 mi Orlando 213 mi 172 mi West Palm each 95 (x 11) aytona each 28. onstruction The W-truss is the most widely used of light wood trusses. Identify two pairs of triangles in the truss below that appear to be congruent. (Lesson 5 4) G 222 hapter 5 Triangles and ongruence

38 HPT 5 Test hoose the letter of the description that best matches each term scalene triangle right triangle isosceles triangle acute triangle equilateral triangle equiangular triangle a. b. c. d. e. f. has a right angle all sides are congruent no sides are congruent has a vertex angle all angles are acute all angles are congruent ind the value of each variable x 9. c 76 b c a 66 c Identify each motion as a translation, reflection, or rotation omplete each congruence statement P Q? 15. L M? N? 16. In, identify the included angle for sides and. etermine whether each pair of triangles is congruent by SSS, SS, S, or S. If it is not possible to prove that they are congruent, write not possible Sports The sail for a sailboat looks like a right triangle. If the angle at the top of the sail measures 54, what is the measure of the acute angle at the bottom? hapter 5 Test 223

39 HPT 5 Preparing for Standardized Tests Statistics Problems Test-Taking Tip On some standardized tests, you will calculate the mean, median, and mode of a data set. You will also choose the most appropriate measure for a data set. On the ST and T, you will apply the concept of the mean to solve problems. mean sum of the numbers number of numbers Memory Tip highway median is in the middle of the road. So a median is the middle number of an ordered data set. median middle number of a set arranged in numerical order mode the number(s) that occurs most often xample 1 xample 2 The heights of ten National hampion Trees are listed in the table below. What is the median, in feet, of the heights? Tree merican eech lack Willow oast ouglas ir oast edwood Giant Sequoia Height (ft) Tree Height (ft) 115 Loblolly Pine Pinyon Pine Sugar Maple Sugar Pine White Oak 79 Hint If there is no single middle number, find the median by calculating the mean of the two middle values. Solution To find the median, first list the heights in numerical order Since there are ten numbers, there is no middle number. The two numbers in the middle are 115 and 148. alculate the mean of these two numbers or The median is 1312 feet. 224 hapter 5 Triangles and ongruence If the average of five numbers is 32 and the average of two of the numbers is 20, then what is the sum of the remaining three numbers? Hint Use the formula for mean to calculate the sum of the numbers. Solution On the ST, average is the same as mean. irst find the sum of the five numbers. Then use the formula for the mean. You know the average (32) and the number of numbers (5). 32 sum of the five numbers sum of the five numbers sum of the five numbers Use the same method to find the sum of the two numbers. 20 sum of the two numbers 2 40 sum of the two numbers You can find the sum of the other three numbers by subtracting: (sum of the five numbers) (sum of the two numbers) or 120. The answer is.

40 Preparing for Standardized Tests or test-taking strategies and more practice, see pages fter you work each problem, record your answer on the answer sheet provided or on a sheet of paper. Multiple hoice 1. Mr. Mendosa obtained estimates for painting from five companies. The estimates were $950, $850, $995, $1000, and $950. What is the mode of these estimates? (Statistics eview) $150 $949 $950 $ Jenny recorded high temperatures every day for a week. The temperatures, in degrees ahrenheit, were 48, 55, 60, 55, 52, 47, and 40. What was the mean temperature? (Statistics eview) What is the value of x in the figure? (Lesson 5 2) 1.5x ? (lgebra eview) x 5x Jared s study group recorded the time they spent on math homework one day. Here are the results (in minutes): 30, 29, 32, 25, 36, 20, 30, 26, 56, 45, 33, and 34. What was the median time spent? (Statistics eview) 20 min 25 min 30 min 31 min 4. The figure below shows an example of a (Lesson 5 3) Grid In 1 9. There are 24 fish in an aquarium. If 8 of 2 them are tetras and 3 of the remaining fish are guppies, how many guppies are there? (lgebra eview) xtended esponse 10. The table shows the percent of new passenger cars imported into the United States by country of origin in (Statistics eview) Percent of New Passenger ars Imported into U.S. by ountry of Origin ountry New ars (percent) dilation. rotation. reflection. translation. 5. Yoshi wants to buy a sweater priced at $ If the sales tax rate is 6%, which is the best estimate of the tax paid on the sweater? (Percent eview) $3.00 $3.60 $4.00 $ How many even integers are there between 2 and 100, not including 2 and 100? (lgebra eview) anada Germany Japan Mexico Korea Other Source: ureau of ensus, oreign Trade ivision Part Make a circle graph to show the data. Label each section of the graph with the percent of imported cars. Part The total value of cars imported was about $114 billion. Use this information to determine the value of cars imported from outside North merica. hapter 5 Preparing for Standardized Tests 225