SAS Triangle Congruence
|
|
- Jane Matthews
- 5 years ago
- Views:
Transcription
1 OMMON OR Locker LSSON 5.3 SS Triangle ongruence Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? ommon ore Math Standards The student is expected to: OMMON OR G-O..8 xplain how the criteria for triangle congruence ( SS...) follow from the definition of congruence in terms of rigid motions. lso G-O..7, G-O..10, G-SRT..5 Mathematical Practices OMMON OR MP.3 Logic Language Objective Have students work in pairs to find an example in the lesson and write out a step-by-step explanation of how the SS Triangle ongruence Theorem works. NGG ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. PRVIW: LSSON PRORMN TSK View the ngage section online. xplain that triangle congruence is important in the design of structures like pyramids. Then preview the Lesson Houghton Mifflin Harcourt Publishing ompany xplore 1 rawing Triangles Given Two Sides and an ngle You know that when all corresponding parts of two triangles are congruent, then the triangles are congruent. Sometimes you can determine that triangles are congruent based on less information. or this activity, cut two thin strips of paper, one long and the other long. On a sheet of paper use a straightedge to draw a horizontal line. rrange the strip to form a angle, as shown. Next, arrange the strip to complete the triangle. How many different triangles can you form? Support your answer with a diagram. 2 different triangles Now arrange the two strips of paper to form a angle so that the angle is included between the two consecutive sides, as shown. With this arrangement, can you construct more than one triangle? Why or why not? No, only one triangle is possible. Having the angle included between the sides fixes the position of the sides. Resource Locker Performance Task. 245 Module Lesson 3 Name lass ate 5.3 SS Triangle ongruence ssential Question: What does the SS Triangle ongruence Theorem tell you about triangles? xplore 1 rawing Triangles Given Two Sides and an ngle You know that when all corresponding parts of two triangles are congruent, then the triangles are congruent. Sometimes you can determine that triangles are congruent based on less information. or this activity, cut two thin strips of paper, one long and the other long. Houghton Mifflin Harcourt Publishing ompany G-O..8 xplain how the criteria for triangle congruence ( SS ) follow from the definition of congruence in terms of rigid motions. lso G-O..7, G-O..10, G-SRT..5 On a sheet of paper use a straightedge to draw a horizontal line. rrange the strip to form a angle, as shown. Next, arrange the strip to complete the triangle. How many different triangles can you form? Support your answer with a diagram. 2 different triangles Now arrange the two strips of paper to form a angle so that the angle is included between the two consecutive sides, as shown. With this arrangement, can you construct more than one triangle? Why or why not? No, only one triangle is possible. Having the angle included between the sides fixes the position of the sides. Resource HROVR PGS Turn to these pages to find this lesson in the hardcover student edition. Module Lesson Lesson 5.3
2 Reflect 1. iscussion If two triangles have two pairs of congruent corresponding sides and one pair of congruent corresponding angles, under what conditions can you conclude that the triangles must be congruent? xplain. The triangles must be congruent if the congruent corresponding angles are the angles included between the congruent corresponding sides. XPLOR 1 rawing Triangles Given Two Sides and an ngle xplore 2 Justifying SS Triangle ongruence You can explain the results of xplore 1 using transformations. onstruct by copying, side, and side. Let point correspond to point, point correspond to point, and point correspond to point, and place point on the segment shown. The diagram illustrates one step in a sequence of rigid motions that will map onto. escribe a complete sequence of rigid motions that will map onto. What can you conclude about the relationship between and? xplain your reasoning. because there is a sequence of rigid motions that maps one onto the other. Reflect 2. Is it possible to map onto using a single rigid motion? If so, describe the rigid motion. Yes; possible answer: reflect across a vertical line halfway between points and. Possible answer: Translate so that point maps to point. Then rotate 180 counterclockwise about point. Point will map to point because =. Then reflect across. Point will map to point because and =. Module Lesson 3 Houghton Mifflin Harcourt Publishing ompany INTGRT THNOLOGY Have students use geometry software to explore included angles. QUSTIONING STRTGIS If I hold up a compass and increase the angle, what happens to the distance between the tips? If I decrease the angle, what happens to the distance between the tips? If I keep the angle the same, what happens to the distance between the tips? The angle increases; it decreases; it stays the same. XPLOR 2 Justifying SS Triangle ongruence INTGRT MTHMTIL PRTIS ocus on ritical Thinking MP.3 ach time the students perform a transformation, have them note the effect of the transformation on the angles and sides. They should notice that they are transformed in the same way and that their measures stay the same. PROSSIONL VLOPMNT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.3, which calls for students to construct viable arguments and critique the reasoning of others. s students explore congruent triangles, ask them to share their observations and conclusions with the class. s they share their findings, ask if anyone got different results. iscuss the differences. Promoting this type of dialogue in the classroom is an essential aspect of the standard. QUSTIONING STRTGIS oes it matter on which side of the angle you place each segment? No; they will make the same triangle, with the only difference being a reflection. SS Triangle ongruence 246
3 XPLIN 1 eciding Whether Triangles re ongruent Using SS Triangle ongruence xplain 1 eciding Whether Triangles are ongruent Using SS Triangle ongruence What you explored in the previous two activities can be summarized in a theorem. You can use this theorem and the definition of congruence in terms of rigid motions to determine whether two triangles are congruent. SS Triangle ongruence Theorem If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. QUSTIONING STRTGIS How do you know that two sides of a triangle are congruent? Two sides are congruent if they have the same length. INTGRT MTHMTIL PRTIS ocus on ritical Thinking MP.3 Remind students that they know how to find the measure of an angle of a triangle when they know the measures of the other two angles. This makes it possible to apply the S Triangle ongruence Theorem to many sets of triangles. Tell them to suppose that they know the lengths of two sides of a triangle. Is it possible to use that information to find the length of the third side? It is possible only if the triangle is a right triangle; then the length of the third side can be found using the Pythagorean Theorem. Houghton Mifflin Harcourt Publishing ompany xample 1 etermine whether the triangles are congruent. xplain your reasoning. Look for congruent corresponding parts. Sides and do not correspond to side 20 cm, 43 because they are not 15 cm long. corresponds to 19 cm 15 cm, because = = 20 cm. corresponds to, because = = 19 cm. and are corresponding angles because they are included between pairs of corresponding sides, but they don t have the same measure. The triangles are not congruent, because there is no sequence of rigid motions that maps onto. 74 in. J in. L K N P 46 in. 37 M 74 in. Look for congruent corresponding parts. JL corresponds to MP, because JL = MP = 46 in. JK corresponds to MN, because JK = MN = 74 in. J corresponds to M, because m J = m M = 37. Two sides and the included angle of JKL are congruent to two sides and the included angle of MNP. JKL MNP SS Triangle ongruence Theorem by the. 20 cm 19 cm Module Lesson 3 OLLORTIV LRNING Small Group ctivity Instruct students to illustrate the difference between the S and SS Triangle ongruence Theorems. They may make a poster, write an essay, create a model, or use another technique to convey the information. Have students share their work in small groups. Then have each group choose one project to present to the class. 247 Lesson 5.3
4 Your Turn 3. etermine whether the triangles are congruent. xplain your reasoning. GH, GJ, and G, and and G are included by congruent corresponding sides. HGJ by the SS Triangle ongruence Theorem. H 2.5 cm 2.5 cm 2.9 cm G 1.7 cm 1.7 cm XPLIN 2 Proving Triangles re ongruent Using SS Triangle ongruence xplain 2 Proving Triangles re ongruent Using SS Triangle ongruence Theorems about congruent triangles can be used to show that triangles in real-world objects are congruent. xample 2 Write each proof. J VOI OMMON RRORS Remind students that they should not assume information from a figure unless it is marked or stated in the given information. Write a proof to show that the two halves of a triangular window are congruent if the vertical post is the perpendicular bisector of the base. Given: is the perpendicular bisector of. Prove: It is given that is the perpendicular bisector of. y the definition of a perpendicular bisector, =, which means, and, which means and are congruent right angles. In addition, by the reflexive property of congruence. So two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. Given: bisects and bisects Prove: It is given that bisects and bisects. So by the definition = of a bisector, = and, which makes and. because they are vertical angles included. So two sides and the angle of are congruent to two sides and the included angle of. The SS Triangle ongruence Theorem triangles are congruent by the. Houghton Mifflin Harcourt Publishing ompany Image redits: Ulrich Niehoff/Imagebroker/age fotostock QUSTIONING STRTGIS To use SS, is it essential that the congruent angles be included between the pairs of congruent sides? Yes, because it is possible for an acute triangle and an obtuse triangle to have two pairs of corresponding congruent sides and a pair of corresponding congruent nonincluded angles. There is no Side-Side-ngle (SS) Theorem. Module Lesson 3 IRNTIT INSTRUTION Kinesthetic xperience Have students place two pencils on their desks so that the points intersect and the pencils model an angle. Have students measure the distance between the erasers. Have the students rotate one pencil to change the angle. Have them measure the distance between the erasers again. fter they have experimented with different angles, discuss whether or not it is possible to change the distance between the erasers without changing the angle. SS Triangle ongruence 248
5 LORT QUSTIONING STRTGIS The solution to an exercise is JKL MNP. Suppose that ud concludes that JLK MPN and Kim concludes that KJL MPN. an both students be correct? xplain. ud s answer is correct because the order of the vertices lines up congruent angles. Kim s is not because the order of the vertices does not line up congruent angles. Your Turn 4. Given: and 1 2 Prove: Possible answer: You are given that and 1 2. You also know that by the reflexive property. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem laborate 5. xplain why the corresponding angles must be included angles in order to use the SS Triangle ongruence Theorem. Possible answer: If the corresponding angles are not included angles, then there is more than one possible angle between the congruent corresponding sides. 1 2 SUMMRIZ TH LSSON Why would you use the SS Triangle ongruence Theorem? What do you need to know to use it? You would use the SS Triangle ongruence Theorem to prove that two triangles are congruent by using only three pairs of congruent parts. You need to know that two pairs of corresponding sides are congruent and the angles included between those sides are also congruent. Houghton Mifflin Harcourt Publishing ompany 6. Jeffrey draws PQR and TUV. He uses a translation to map point P to point T and point R to point V as shown. What should be his next step in showing the triangles are congruent? Why? Reflect PQR across TV ; this will map point Q to point U and show that there is a sequence of rigid motions that maps PQR to TUV. 7. ssential Question heck-in If two triangles share a common side, what else must be true for the SS Triangle ongruence Theorem to apply? second side and an included angle must be congruent. valuate: Homework and Practice 1. Sarah performs rigid motions mapping point to point and point to point, as shown. oes she have enough information to confirm that the triangles are congruent? xplain your reasoning. No; she can map to by a reflection across, but will map to only if =. Q P T U R V Online Homework Hints and Help xtra Practice Module Lesson 3 LNGUG SUPPORT onnect Vocabulary Open and shut a door and talk about the function of a hinge. ompare the concept of an included angle to a hinge. raw a triangle on the board, labeling the vertices. Have students identify the angle that is included between each pair of sides. 249 Lesson 5.3
6 etermine whether the triangles are congruent. xplain your reasoning Q P 2 in. S 2 in. PS RQ, PR PR, SPR QRP, and SPR and QRP are included by congruent corresponding sides. SPR QRP by SS H 30 mm 40 mm 30 mm mm G J 52 mm GH and HJ, but included angles and H are not congruent. The triangles are not congruent, because there is no sequence of rigid motions that maps onto GHJ. ind the value of the variable that results in congruent triangles. xplain in 30 (2x) in. R 9 in 2x = x + 4; x = 4; by SS when x is (x + 4) in. 30 mm 30 mm,, and, and and are included by congruent corresponding sides. by SS. 1.3 m 1.3 m,, and and are included by congruent corresponding sides. by SS. (2x + 14) 13.5 in. 12 in in. (4x) 12 in. 2x+ 14 = 4x; x = 7; by SS when x is 7. Houghton Mifflin Harcourt Publishing ompany VLUT SSIGNMNT GUI oncepts and Skills xplore 2 Justifying SS Triangle ongruence xample 1 eciding Whether Triangles re ongruent Using SS Triangle ongruence xample 2 Proving Triangles re ongruent Using SS Triangle ongruence Practice xercises 1 6 xercises 7 11 xercises INTGRT MTHMTIL PRTIS ocus on ommunication MP.3 Write S and SS on the board. sk students to do each of the following in their Math Journals: Tell what each stands for in terms of triangle congruence. raw and label a diagram illustrating each. Tell how the two theorems are the same. Tell how the two theorems are different. Module Lesson 3 xercise epth of Knowledge (.O.K.) OMMON OR Mathematical Practices Skills/oncepts MP.3 Logic 8 2 Skills/oncepts MP.1 Problem Solving Skills/oncepts MP.4 Modeling Skills/oncepts MP.3 Logic 13 3 Strategic Thinking MP.4 Modeling 14 3 Strategic Thinking MP.2 Reasoning 15 3 Strategic Thinking MP.6 Precision SS Triangle ongruence 250
7 RITIL THINKING raw non-collinear points M, O, and U on the board, connecting them to form an obtuse angle with vertex O. sk students to visualize a translation, a rotation, and a reflection of the figure shown. In each case, have them describe the effect on the segment that connects point M with point U. Sample response: The segment connecting points M and U will follow the same movements as the rest of the figure. Its length will remain the same no matter what rigid-motion transformation is used. 8. Given that polygon is a regular hexagon, prove that. Statements Reasons 1. is a regular hexagon. 1. Given 2. = and = 2. efinition of regular polygon 3. and 3. efinition of congruence in terms of rigid motion 4. m = m 4. efinition of regular polygon efinition of congruence in terms of rigid motion SS Triangle ongruence Theorem PT 9. product designer is designing an easel with extra braces as shown in the diagram. Prove that if and, then the braces and are also congruent. Houghton Mifflin Harcourt Publishing ompany Image redits: ndreyuu/istockphoto.com You are given that and. You also know that by the reflexive property. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. So, by PT, the braces and and are also congruent. Module Lesson Lesson 5.3
8 10. n artist is framing a large picture and wants to put metal poles across the back to strengthen the frame as shown in the diagram. If the metal poles are both the same length and they bisect each other, prove that and. VOI OMMON RRORS Students may choose the wrong angle when SS is used to prove triangles congruent. xplain that the angle must be formed by the sides. The included angle is named by the letter the segments share. ecause and bisect each other, = and =, so and by the definition of congruence. y the Vertical ngle Theorem, you also know that. Two sides and the included angle of are congruent to two sides and the included angle of. The triangles are congruent by the SS Triangle ongruence Theorem. y PT,. You can use similar reasoning to show that 11. The figure shows a side panel of a skateboard ramp. Kalim wants to confirm that the right triangles in the panel are congruent. a. What measurements should Kalim take if he wants to confirm that the triangles are congruent by SS? xplain. Measure and ; so he can confirm that two pairs of sides and their included angles are congruent. (,, and ) b. What measurements should Kalim take if he wants to confirm that the triangles are congruent by S? xplain. Measure and ; so he can confirm that two pairs of angles and their included sides are congruent. (,, and ) Houghton Mifflin Harcourt Publishing ompany Module Lesson 3 SS Triangle ongruence 252
9 JOURNL Have students describe how they can use color coding to help them recognize the SS theorem. Have them support their descriptions with colored sketches. 12. Which of the following are reasons that justify why the triangles are congruent? Select all that apply.. SS Triangle ongruence Theorem. SS Triangle ongruence Theorem. S Triangle ongruence Theorem. onverse of PT. PT. SS is not a valid congruence theorem.. You do not know that all of the corresponding parts are congruent.. PT is a property of congruent triangles, not a justification for congruence. H.O.T. ocus on Higher Order Thinking 13. Multi-Step Refer to the following diagram to answer each question. Houghton Mifflin Harcourt Publishing ompany a. Use a triangle congruence theorem to explain why these triangles are congruent. ach triangle has side lengths of 2 and 6 and an included right angle. y SS they are congruent. b. escribe a sequence of rigid motions to map the top triangle onto the bottom triangle to confirm that they are congruent. Possible answer: Reflect the triangle across the y-axis. Next translate it 1 unit to the left. Then translate it 6 units down. Module Lesson Lesson 5.3
10 14. xplain the rror Mark says that the diagram confirms that a given angle and two given side lengths determine a unique triangle even if the angle is not an included angle. xplain Mark s error. 15. Justify Reasoning The opposite sides of a rectangle are congruent. an you conclude that a diagonal of a rectangle divides the rectangle into two congruent triangles? Justify your response. Lesson Performance Task J The diagram of the Great Pyramid at Giza gives the approximate lengths of edge and slant height. The slant height is the perpendicular bisector of. ind the perimeter of. xplain how you found the answer. ecause is the perpendicular bisector of, = and m = m = 90. lso, = so by the SS Triangle ongruence Theorem. Therefore, = 720 ft by PT. To find, use the Pythagorean Theorem: () 2 = = 158,400; = 158, ; =, so = 398 perimeter of = = 2,236 ft. L M Possible answer: circle with its center at M and radius MJ = MK will intersect JL and KL at two other points closer to L. The triangles formed by each of these two points and the points L and M will be different than the original triangles, even though they are formed by the same given angle and two given side lengths. Yes; since the opposite sides of a rectangle are congruent and the included angles between the sides are right angles, the two triangles are congruent by the SS Theorem. K edge 720 ft slant height 600 ft Houghton Mifflin Harcourt Publishing ompany INTGRT MTHMTIL PRTIS ocus on Reasoning MP.2 sk students to identify the single congruence they would need to establish, in addition to the given information, to enable them to prove by the S Triangle ongruence Theorem. INTGRT MTHMTIL PRTIS ocus on Math onnections MP.1 The base of the Great Pyramid is square. What is the area of the base in acres? (1 acre = 43,560 ft 2 ) about 14.5 acres Module Lesson 3 XTNSION TIVITY Some authorities claim that there is a relationship between the dimensions of the Great Pyramid at Giza and π, the ratio of the circumference of a circle to the radius. Have students research this claim and offer evidence as to its truth or falsity. The claim is that the ratio of the perimeter of the base of the Great Pyramid to its height equals 2π. Some sources give 923 m as the perimeter of the base and m as the height ; 2π Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. SS Triangle ongruence 254
SAS Triangle Congruence
Locker LSSON 5.3 SS Triangle ongruence Texas Math Standards The student is expected to: G.6. Prove two triangles are congruent by applying the Side-ngle-Side, ngle-side-ngle, Side-Side-Side, ngle-ngle-side,
More informationASA Triangle Congruence
Locker LSSON 5.2 S Triangle ongruence Texas Math Standards The student is expected to: G.6. Prove two triangles are congruent by applying the Side-ngle-Side, ngle-side-ngle, Side-Side-Side, ngle-ngle-side,
More information5.4 SSS Triangle Congruence
OMMON OR Locker LSSON ommon ore Math Standards The student is expected to: OMMON OR G-O..8 xplain how the criteria for triangle congruence (... SSS) follow from the definition of congruence in terms of
More informationEssential Question: What does the AAS Triangle Congruence Theorem tell you about two triangles? Explore Exploring Angle-Angle-Side A C E F
OMMON OR G G Locker LSSON 6. S Triangle ongruence Name lass ate 6. S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? ommon ore Math Standards
More informationCorresponding Parts of Congruent Figures Are Congruent
OMMON OR Locker LSSON 3.3 orresponding arts of ongruent igures re ongruent Name lass ate 3.3 orresponding arts of ongruent igures re ongruent ssential Question: What can you conclude about two figures
More information5.2 ASA Triangle Congruence
Name lass ate 5.2 S Triangle ongruence ssential question: What does the S Triangle ongruence Theorem tell you about triangles? xplore 1 rawing Triangles Given Two ngles and a Side You have seen that two
More information11.4 AA Similarity of Triangles
Name lass ate 11.4 Similarity of Triangles ssential Question: How can you show that two triangles are similar? xplore G.7. pply the ngle-ngle criterion to verify similar triangles and apply the proportionality
More information11.4 AA Similarity of Triangles
Name lass ate 11.4 Similarity of Triangles ssential Question: How can you show that two triangles are similar? xplore xploring ngle-ngle Similarity for Triangles Two triangles are similar when their corresponding
More information5.4 SSS Triangle Congruence
Locker LSSON 5.4 SSS Triangle ongruence Name lass ate 5.4 SSS Triangle ongruence ssential uestion: What does the SSS Triangle ongruence Theorem tell you about triangles? Texas Math Standards The student
More information15.2 Angles in Inscribed Quadrilaterals
Name lass ate 15.2 ngles in Inscribed Quadrilaterals Essential Question: What can you conclude about the angles of a quadrilateral inscribed in a circle? Resource Locker Explore Investigating Inscribed
More information7.2 Isosceles and Equilateral Triangles
Name lass Date 7.2 Isosceles and Equilateral Triangles Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles? Resource Locker Explore G.6.D
More informationIsosceles and Equilateral Triangles
OMMON ORE Locker LESSON ommon ore Math Standards The student is expected to: OMMON ORE G-O..10 Prove theorems about triangles. Mathematical Practices OMMON ORE 7.2 Isosceles and Equilateral Triangles MP.3
More information9.2 Conditions for Parallelograms
Name lass ate 9.2 onditions for Parallelograms Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram? Explore G.6.E Prove a quadrilateral is a parallelogram...
More information6.2 AAS Triangle Congruence
Name lass ate 6. S Triangle ongruence ssential Question: What does the S Triangle ongruence Theorem tell ou about two triangles? xplore G.6. Prove two triangles are congruent b appling the ngle-ngle-side
More informationAngle Theorems for Triangles 8.8.D
? LSSON 7.2 ngle Theorems for Triangles SSNTIL QUSTION xpressions, equations, and relationships 8.8. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, What
More information24.2 Conditions for Parallelograms
Locker LESSON 24 2 onditions for Parallelograms Name lass ate 242 onditions for Parallelograms Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram? ommon ore
More information9.4 Conditions for Rectangles, Rhombuses, and Squares
Name lass ate 9.4 onditions for Rectangles, Rhombuses, and Squares ssential Question: ow can you use given conditions to show that a quadrilateral is a rectangle, a rhombus, or a square? Resource Locker
More information6.3 HL Triangle Congruence
Name lass ate 6.3 HL Triangle ongruence Essential Question: What does the HL Triangle ongruence Theorem tell you about two triangles? Explore Is There a Side-Side-ngle ongruence Theorem? Resource Locker
More information20.1 Exploring What Makes Triangles Congruent
ame lass ate 20.1 xploring What akes Triangles ongruent ssential Question: How can you show that two triangles are congruent? esource ocker xplore Transforming Triangles with ongruent orresponding arts
More information14.2 Angles in Inscribed Quadrilaterals
Name lass ate 14.2 ngles in Inscribed Quadrilaterals Essential Question: What can you conclude about the angles of a quadrilateral inscribed in a circle? Explore G.12. pply theorems about circles, including
More informationCongruence Transformations and Triangle Congruence
ongruence Transformations and Triangle ongruence Truss Your Judgment Lesson 11-1 ongruent Triangles Learning Targets: Use the fact that congruent triangles have congruent corresponding parts. etermine
More informationEssential Question How can you prove that a quadrilateral is a parallelogram? Work with a partner. Use dynamic geometry software.
OMMON OR Learning Standards HSG-O..11 HSG-SRT..5 HSG-MG..1 RSONING STRTLY 7.3 To be proficient in math, you need to know and flexibly use different properties of objects. Proving That a Quadrilateral Is
More informationName: Unit 4 Congruency and Triangle Proofs
Name: Unit 4 ongruency and Triangle Proofs 1 2 Triangle ongruence and Rigid Transformations In the diagram at the right, a transformation has occurred on. escribe a transformation that created image from.
More informationIsosceles and Equilateral Triangles
Locker LESSON 7.2 Isosceles and Equilateral Triangles Texas Math Standards The student is expected to: G.6.D Verify theorems about the relationships in triangles, including... base angles of isosceles
More information9.3 Properties of Rectangles, Rhombuses, and Squares
Name lass Date 9.3 Properties of Rectangles, Rhombuses, and Squares Essential Question: What are the properties of rectangles, rhombuses, and squares? Resource Locker Explore Exploring Sides, ngles, and
More information18.2 Sine and Cosine Ratios
Name lass ate 18.2 Sine and osine Ratios ssential Question: How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles? Resource Locker xplore Investigating
More information6 segment from vertex A to BC. . Label the endpoint D. is an altitude of ABC. 4 b. Construct the altitudes to the other two sides of ABC.
6. Medians and ltitudes of Triangles ssential uestion What conjectures can you make about the medians and altitudes of a triangle? inding roperties of the Medians of a Triangle Work with a partner. Use
More informationKey Concept Congruent Figures
4-1 ongruent igures ommon ore State Standards Prepares for G-SRT..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective
More informationMaintaining Mathematical Proficiency
Name ate hapter 5 Maintaining Mathematical Proficiency Find the coordinates of the midpoint M of the segment with the given endpoints. Then find the distance between the two points. 1. ( 3, 1 ) and ( 5,
More information7.3 Triangle Inequalities
Name lass Date 7.3 Triangle Inequalities Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? Resource Locker Explore Exploring
More informationb. Move BC so that B is on the smaller circle and C is on the larger circle. Then draw ABC.
5.5 Proving Triangle ongruence by ssential uestion What can you conclude about two triangles when you know the corresponding sides are congruent? rawing Triangles Work with a partner. Use dynamic geometry
More informationGeometry Definitions, Postulates, and Theorems. Chapter 4: Congruent Triangles. Section 4.1: Apply Triangle Sum Properties
Geometry efinitions, Postulates, and Theorems Key hapter 4: ongruent Triangles Section 4.1: pply Triangle Sum Properties Standards: 12.0 Students find and use measures of sides and of interior and exterior
More information1.1 Segment Length and Midpoints
Name lass ate 1.1 Segment Length and Midpoints Essential Question: How do you draw a segment and measure its length? Explore Exploring asic Geometric Terms In geometry, some of the names of figures and
More informationTo use and apply properties of isosceles and equilateral triangles
- Isosceles and Equilateral riangles ontent Standards G.O. Prove theorems about triangles... base angles of isosceles triangles are congruent... lso G.O., G.SR. Objective o use and apply properties of
More informationMaintaining Mathematical Proficiency
Name ate hapter 6 Maintaining Mathematical Proficiency Write an equation of the line passing through point P that is perpendicular to the given line. 1. P(5, ), y = x + 6. P(4, ), y = 6x 3 3. P( 1, ),
More information3.3 Corresponding Parts of Congruent Figures Are Congruent
Name lass ate 3.3 orresponding arts of ongruent Figures re ongruent Essential Question: What can you conclude about two figures that are congruent? esource Locker Explore G.6. pply the definition of congruence,
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 5: Congruent Triangles Instruction
Prerequisite Skills This lesson requires the use of the following skills: understanding that rigid motions maintain the shape and size of angles and segments, and that rigid motions include the transformations
More informationb) A ray starts at one point on a line and goes on forever. c) The intersection of 2 planes is one line d) Any four points are collinear.
Name: Review for inal 2016 Period: eometry 22 Note to student: This packet should be used as practice for the eometry 22 final exam. This should not be the only tool that you use to prepare yourself for
More information10.5 Perimeter and Area on the Coordinate Plane
Name lass ate 1.5 Perimeter and rea on the oordinate Plane ssential Question: How do ou find the perimeter and area of polgons in the coordinate plane? Resource Locker plore inding Perimeters of igures
More informationMaintaining Mathematical Proficiency
Name ate hapter 8 Maintaining Mathematical Proficiency Tell whether the ratios form a proportion. 1. 16, 4 12 2. 5 45, 6 81. 12 16, 96 100 4. 15 75, 24 100 5. 17 2, 68 128 6. 65 156, 105 252 Find the scale
More information7.1 Interior and Exterior Angles
COMMON CORE Locker l LESSON 7. Interior and Exterior ngles Name Class Date 7. Interior and Exterior ngles Essential Question: What can you say about the interior and exterior angles of a triangle and other
More information5.4. Equilateral and Isosceles Triangles
OMMON OR Learning Standards HSG-O..10 HSG-O..13 HSG-MG..1.4 ONSRUING VIL RGUMNS o be proficient in math, you need to make conjectures and build a logical progression of statements to explore the truth
More information4-2 Triangle Congruence Conditions. Congruent Triangles - C F. and
4-2 Triangle ongruence onditions ongruent Triangles -,, ª is congruent to ª (ª ª) under a correspondence of parts if and only if 1) all three pairs of corresponding angles are congruent, and 2) all three
More information2.7 Angles and Intersecting Lines
Investigating g Geometry TIVITY Use before Lesson.7.7 ngles and Intersecting Lines M T R I LS graphing calculator or computer Q U S T I O N What is the relationship between the measures of the angles formed
More informationFinal Exam Review. Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the length of.
Final Exam Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the length of. 9 8 7 6 5 4 3 2 1 0 1 a. = 7 c. = 7 b. = 9 d. = 8 2. Find the best
More informationCHAPTER # 4 CONGRUENT TRIANGLES
HPTER # 4 ONGRUENT TRINGLES In this chapter we address three ig IES: 1) lassify triangles by sides and angles 2) Prove that triangles are congruent 3) Use coordinate geometry to investigate triangle relationships
More informationTo recognize congruent figures and their corresponding parts
4-1 ongruent igures ontent Standard Prepares for G.SR.5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. Objective o recognize congruent figures
More informationDO NOT LOSE THIS REVIEW! You will not be given another copy.
Geometry Fall Semester Review 2011 Name: O NOT LOS THIS RVIW! You will not be given another copy. The answers will be posted on your teacher s website and on the classroom walls. lso, review the vocabulary
More informationGeometry. Transformations. Slide 1 / 273 Slide 2 / 273. Slide 4 / 273. Slide 3 / 273. Slide 5 / 273. Slide 6 / 273.
Slide 1 / 273 Slide 2 / 273 Geometry Transformations 2015-10-26 www.njctl.org Slide 3 / 273 Slide 4 / 273 Table of ontents Transformations Translations Reflections Rotations Identifying Symmetry with Transformations
More informationMy Notes. Activity 31 Quadrilaterals and Their Properties 529
Quadrilaterals and Their Properties 4-gon Hypothesis Learning Targets: Develop properties of kites. Prove the Triangle idsegment Theorem. SUGGSTD LRNING STRTGIS: reate Representations, Think-Pair-Share,
More information18.1 Sequences of Transformations
Name lass ate 1.1 Sequences of Transformations ssential Question: What happens when ou appl more than one transformation to a figure? plore ombining Rotations or Reflections transformation is a function
More information3.2 Proving Figures are Congruent Using Rigid Motions
Name lass ate 3.2 Proving igures are ongruent Using igid otions ssential uestion: How can ou determine whether two figures are congruent? esource ocker plore onfirming ongruence Two plane figures are congruent
More informationGeometry Unit 4a - Notes Triangle Relationships
Geometry Unit 4a - Notes Triangle Relationships This unit is broken into two parts, 4a & 4b. test should be given following each part. Triangle - a figure formed by three segments joining three noncollinear
More informationParallel Lines and Triangles. Objectives To use parallel lines to prove a theorem about triangles To find measures of angles of triangles
-5 Parallel Lines and Triangles ommon ore State Standards G-O..0 Prove theorems about triangles... measures of interior angles of a triangle sum to 80. MP, MP, MP 6 Objectives To use parallel lines to
More informationMath 366 Chapter 12 Review Problems
hapter 12 Math 366 hapter 12 Review Problems 1. ach of the following figures contains at least one pair of congruent triangles. Identify them and tell why they are congruent. a. b. G F c. d. e. f. 1 hapter
More informationUNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 7: Proving Similarity Instruction
Prerequisite Skills This lesson requires the use of the following skills: creating ratios solving proportions identifying both corresponding and congruent parts of triangles Introduction There are many
More informationTo prove two triangles congruent using the SSS and SAS Postulates. Are the triangles below congruent? How do you know? 6 B 4
4-2 riangle ongruence by SSS and SS ommon ore State Standards -SR..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective
More information7.3 Triangle Inequalities
Locker LESSON 7.3 Triangle Inequalities Name lass Date 7.3 Triangle Inequalities Teas Math Standards The student is epected to: G.5.D Verify the Triangle Inequality theorem using constructions and apply
More information4.0 independently go beyond the classroom to design a real-world connection with polygons that represents a situation in its context.
ANDERSON Lesson plans!!! Intro to Polygons 10.17.16 to 11.4.16 Level SCALE Intro to Polygons Evidence 4.0 independently go beyond the classroom to design a real-world connection with polygons that represents
More informationFirst Nations people use a drying rack to dry fish and animal hides. The drying rack in this picture is used in a Grade 2 classroom to dry artwork.
7.1 ngle roperties of Intersecting Lines Focus Identify and calculate complementary, supplementary, and opposite angles. First Nations people use a drying rack to dry fish and animal hides. The drying
More informationFirst published in 2013 by the University of Utah in association with the Utah State Office of Education.
First published in 013 by the University of Utah in association with the Utah State Office of Education. opyright 013, Utah State Office of Education. Some rights reserved. This work is published under
More informationProperties of Rhombuses, Rectangles, and Squares
6- Properties of Rhombuses, Rectangles, and Squares ontent Standards G.O. Prove theorems about parallelograms... rectangles are parallelograms with congruent diagonals. lso G.SRT.5 Objectives To define
More informationGeo Final Review 2014
Period: ate: Geo Final Review 2014 Multiple hoice Identify the choice that best completes the statement or answers the question. 1. n angle measures 2 degrees more than 3 times its complement. Find the
More informationParallel Lines Cut by a Transversal. ESSENTIAL QUESTION What can you conclude about the angles formed by parallel lines that are cut by a transversal?
? LSSON 7.1 Parallel Lines ut by a Transversal SSNTIL QUSTION What can you conclude about the angles formed by parallel lines that are cut by a transversal? xpressions, equations, and relationships Use
More informationSlide 1 / 343 Slide 2 / 343
Slide 1 / 343 Slide 2 / 343 Geometry Quadrilaterals 2015-10-27 www.njctl.org Slide 3 / 343 Table of ontents Polygons Properties of Parallelograms Proving Quadrilaterals are Parallelograms Rhombi, Rectangles
More informationDay 1: Geometry Terms & Diagrams CC Geometry Module 1
Name ate ay 1: Geometry Terms & iagrams Geometry Module 1 For #1-3: Identify each of the following diagrams with the correct geometry term. #1-3 Vocab. ank Line Segment Line Ray 1. 2. 3. 4. Explain why
More informationGeometry. Slide 1 / 190 Slide 2 / 190. Slide 4 / 190. Slide 3 / 190. Slide 5 / 190. Slide 5 (Answer) / 190. Angles
Slide 1 / 190 Slide 2 / 190 Geometry ngles 2015-10-21 www.njctl.org Slide 3 / 190 Table of ontents click on the topic to go to that section Slide 4 / 190 Table of ontents for Videos emonstrating onstructions
More informationInvestigating a Ratio in a Right Triangle. Leg opposite. Leg adjacent to A
Name lass ate 13.1 Tangent atio Essential uestion: How do you find the tangent ratio for an acute angle? esource Locker Explore Investigating a atio in a ight Triangle In a given a right triangle,, with
More informationMaintaining Mathematical Proficiency
Name Date Chapter 1 Maintaining Mathematical Proficiency Simplify the expression. 1. 3 + ( 1) = 2. 10 11 = 3. 6 + 8 = 4. 9 ( 1) = 5. 12 ( 8) = 6. 15 7 = + = 8. 5 ( 15) 7. 12 3 + = 9. 1 12 = Find the area
More informationUNIT 5 Measurement Geometry
UNIT 5 Measurement Geometry MOUL MOUL 11 ngle Relationships in Parallel Lines and Triangles MOUL MOUL 12 The Pythagorean Theorem 8.G.2.6, 8.G.2.7, 8.G.2.8 13 Volume MOUL MOUL 8.G.3.9 RRS IN MTH Image redits:
More informationReady to Go On? Skills Intervention 4-1 Classifying Triangles
4 Ready to Go On? Skills Intervention 4-1 lassifying Triangles Find these vocabulary words in Lesson 4-1 and the Multilingual Glossary. Vocabulary acute triangle equiangular triangle right triangle obtuse
More informationEssential Question How can you describe angle pair relationships and use these descriptions to find angle measures?
1.6 escribing Pairs of ngles OMMON OR Learning Standard HSG-O..1 ssential Question How can you describe angle pair relationships and use these descriptions to find angle measures? Finding ngle Measures
More informationGeometry. Points, Lines, Planes & Angles. Part 2. Angles. Slide 1 / 185 Slide 2 / 185. Slide 4 / 185. Slide 3 / 185. Slide 5 / 185.
Slide 1 / 185 Slide 2 / 185 eometry Points, ines, Planes & ngles Part 2 2014-09-20 www.njctl.org Part 1 Introduction to eometry Slide 3 / 185 Table of ontents Points and ines Planes ongruence, istance
More information7.4. Properties of Special Parallelograms For use with Exploration 7.4. Essential Question What are the properties of the diagonals of
Name ate 7.4 Properties of Special Parallelograms For use with xploration 7.4 ssential Question What are the properties of the diagonals of rectangles, rhombuses, and squares? 1 XPLORTION: Identifying
More informationD AC BC AB BD m ACB m BCD. g. Look for a pattern of the measures in your table. Then write a conjecture that summarizes your observations.
OMMON O Learning tandard HG-O..0 6.6 Inequalities in Two Triangles ssential Question If two sides of one triangle are congruent to two sides of another triangle, what can you say about the third sides
More informationGeometry. Transformations. Slide 1 / 154 Slide 2 / 154. Slide 4 / 154. Slide 3 / 154. Slide 6 / 154. Slide 5 / 154. Transformations.
Slide 1 / 154 Slide 2 / 154 Geometry Transformations 2014-09-08 www.njctl.org Slide 3 / 154 Slide 4 / 154 Table of ontents click on the topic to go to that section Transformations Translations Reflections
More information(Current Re nweb Grade)x.90 + ( finalexam grade) x.10 = semester grade
2//2 5:7 PM Name ate Period This is your semester exam which is worth 0% of your semester grade. You can determine grade what-ifs by using the equation below. (urrent Re nweb Grade)x.90 + ( finalexam grade)
More informationMathematics Standards for High School Geometry
Mathematics Standards for High School Geometry Geometry is a course required for graduation and course is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout
More information13.2 Sine and Cosine Ratios
Name lass Date 13.2 Sine and osine Ratios Essential Question: How can you use the sine and cosine ratios, and their inverses, in calculations involving right triangles? Explore G.9. Determine the lengths
More informationUnit 6. Patterns in Shape PRACTICE PROBLEMS. /36 points. Lesson 1: 2-Dimensional Shapes #6, 7, 8, 9, 10. #11, 12, 13 8 points
Name: Unit 6 Patterns in Shape Lesson 1: 2-Dimensional Shapes PRACTICE PROBLEMS I can discover properties of triangles and quadrilaterals, and apply congruence conditions to reason about shapes. Investigation
More informationUnit 3 Part 2 1. Tell whether the three lengths are the sides of an acute triangle, a right triangle, or an obtuse triangle.
HONORS Geometry Final Exam Review 2 nd Semester Name: Unit 3 Part 2 1. Tell whether the three lengths are the sides of an acute triangle, a right triangle, or an obtuse triangle. a. 8, 11, 12 b. 24, 45,
More informationEssential Question What are the properties of parallelograms?
7. roperties of arallelograms ssential uestion What are the properties of parallelograms? iscovering roperties of arallelograms Work with a partner. Use dynamic geometry software. a. onstruct any parallelogram
More informationSTUDY GUIDE REVIEW Similarity and Transformations. 8 y
MODUL Study Guide Review SSSSMNT ND INTRVNTION ssign or customize module reviews. STUDY GUID RVIW Similarity and Transformations ssential Question: How can you use similarity and transformations to solve
More informationGeometry. Points, Lines, Planes & Angles. Part 2. Slide 1 / 185. Slide 2 / 185. Slide 3 / 185. Table of Contents
Slide 1 / 185 Slide 2 / 185 Geometry Points, Lines, Planes & ngles Part 2 2014-09-20 www.njctl.org Part 1 Introduction to Geometry Table of ontents Points and Lines Planes ongruence, istance and Length
More informationEssential Question What are some properties of trapezoids and kites? Recall the types of quadrilaterals shown below.
7.5 Properties of Trapezoids and ites ssential Question What are some properties of trapezoids and kites? ecall the types of quadrilaterals shown below. Trapezoid Isosceles Trapezoid ite PV I OVI PO To
More informationA calculator, scrap paper, and patty paper may be used. A compass and straightedge is required.
The Geometry and Honors Geometry Semester examination will have the following types of questions: Selected Response Student Produced Response (Grid-in) Short nswer calculator, scrap paper, and patty paper
More informationHoughton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry
Houghton Mifflin Harcourt Geometry 2015 correlated to the New York Common Core Learning Standards for Mathematics Geometry Standards for Mathematical Practice SMP.1 Make sense of problems and persevere
More informationName Class Date. Congruence and Transformations Going Deeper
Name lass ate 4-1 ongruence and Transformations Going eeper ssential question: How can ou use transformations to determine whether figures are congruent? Two figures are congruent if the have the same
More informationESSENTIAL QUESTION How can you determine when two triangles are similar? 8.8.D
? LESSON 7.3 ngle-ngle Similarity ESSENTIL QUESTION How can you determine when two triangles are similar? Expressions, equations, and relationships 8.8.D Use informal arguments to establish facts about
More information2017-ACTM Regional Mathematics Contest
2017-TM Regional Mathematics ontest Geometry nswer each of the multiple-choice questions and mark your answers on that answer sheet provided. When finished with the multiple-choice items, then answer the
More informationLesson 1.1 Building Blocks of Geometry
Lesson 1.1 uilding locks of Geometry For xercises 1 7, complete each statement. S 3 cm. 1. The midpoint of Q is. N S Q. NQ. 3. nother name for NS is.. S is the of SQ. 5. is the midpoint of. 6. NS. 7. nother
More informationUNIT PLAN. Big Idea/Theme: Polygons can be identified, classified, and described.
UNIT PLAN Grade Level: 5 Unit #: 11 Unit Name Geometry Polygons Time: 15 lessons, 18 days Big Idea/Theme: Polygons can be identified, classified, and described. Culminating Assessment: (requirements of
More informationKeY TeRM. perpendicular bisector
.6 Making opies Just as Perfect as the Original! onstructing Perpendicular Lines, Parallel Lines, and Polygons LeARnInG GOALS In this lesson, you will: KeY TeRM perpendicular bisector OnSTRUTIOnS a perpendicular
More informationMaintaining Mathematical Proficiency
Name ate hapter 7 Maintaining Mathematical Proficiency Solve the equation by interpreting the expression in parentheses as a single quantity. 1. 5( 10 x) = 100 2. 6( x + 8) 12 = 48 3. ( x) ( x) 32 + 42
More informationWest Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12
West Windsor-Plainsboro Regional School District Basic Geometry Grades 9-12 Unit 1: Basics of Geometry Content Area: Mathematics Course & Grade Level: Basic Geometry, 9 12 Summary and Rationale This unit
More information1. What is the sum of the measures of the angles in a triangle? Write the proof (Hint: it involves creating a parallel line.)
riangle asics irst: Some basics you should already know. eometry 4.0 1. What is the sum of the measures of the angles in a triangle? Write the proof (Hint: it involves creating a parallel line.) 2. In
More informationAngles Formed by Intersecting Lines
1 3 Locker LESSON 19.1 Angles Formed by Intersecting Lines Common Core Math Standards The student is expected to: G-CO.9 Prove theorems about lines and angles. Mathematical Practices MP.3 Logic Name Class
More information13.4 Problem Solving with Trigonometry
Name lass ate 13.4 Problem Solving with Trigonometr Essential Question: How can ou solve a right triangle? Resource Locker Eplore eriving an rea Formula You can use trigonometr to find the area of a triangle
More informationSegment Length and Midpoints
OON ORE Y l Q Locker LESSON 16.1 Segment Length and idpoints Name lass ate 16.1 Segment Length and idpoints Essential Question: How do you draw a segment and measure its length? ommon ore ath Standards
More informationA calculator and patty paper may be used. A compass and straightedge is required. The formulas below will be provided in the examination booklet.
The Geometry and Honors Geometry Semester examination will have the following types of questions: Selected Response Student Produced Response (Grid-in) Short nswer calculator and patty paper may be used.
More information