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# Teacher Resource Sampler. Prentice Hall. Geometry. MatPac101071GeoResourceSampler.indd 1 – PDF Free Download

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1 Teacher Resource Sampler MatPac101071GeoResourceSampler.indd 1 Prentice Hall Geometry 7/1/10 10:51 AM

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3 Prentice Hall Geometry Contents Student Companion 4 Think About a Plan 8 Practice G 9 Practice K 11 Standardized Test Prep 13 Solve It and Lesson Quiz Transparency 14 Additional Problems 15 Reteaching 17 ELL Support 19 Activity 20 Game 21 Puzzle 22 Enrichment 23 Teaching with TI Technology 24 Chapter Quiz 28 Chapter Test 30 Find the Errors! 32 Performance Tasks 35 Extra Practice 37 Chapter Project 40 Cumulative Review 44 3

4 4-1 Congruent Figures Vocabulary Review 1. Underline the correct word to complete the sentence. A polygon is a two-dimensional figure with two / three or more segments that meet exactly at their endpoints. 2. Cross out the figure(s) that are NOT polygons. Vocabulary Builder congruent (adjective) kahng GROO unt Main Idea: Congruent figures have the same size and shape. Related Word: congruence (noun) Use Your Vocabulary 3. Circle the triangles that appear to be congruent. Write T for true or F for false. 4. Congruent angles have different measures. 5. A prism and its net are congruent figures. Copyright by Pearson Education, Inc. or its affiliates. All Rights Reserved. 6. The corresponding sides of congruent figures have the same measure. Chapter

5 Key Concept Congruent Figures Congruent polygons have congruent corresponding parts their matching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order. 7. Use the figures at the right to complete each congruence statement. A D B F C G ABCD EFGH E H AB > BC > CD > DA > /A > /B > /C > /D > Problem 1 Using Congruent Parts Got It? If kwys O kmkv, what are the congruent corresponding parts? 8. Use the diagram at the right. Draw an arrow from each vertex of the first triangle to the corresponding vertex of the second triangle. W Y S M K V 9. Use the diagram from Exercise 8 to complete each congruence statement. Sides WY > YS > WS > Angles /W > /Y > /S > Problem 2 Finding Congruent Parts Copyright by Pearson Education, Inc. or its affiliates. All Rights Reserved. Got It? Suppose that kwys O kmkv. If m/w 5 62 and m/y 5 35, what is mlv? Explain. Use the congruent triangles at the right. 10. Use the given information to label the triangles. Remember to write corresponding vertices in order. 11. Complete each congruence statement. /W > /Y > /S > 12. Use the Triangle Angle-Sum theorem. m/s 1 m 1 m 5 180, so m/s ( 1 ), or. 13. Complete. Since /S > and m/s 5, m/v 5. W Y 91 Lesson 4-1 5

6 Problem 3 Finding Congruent Triangles Got It? Is kabd O kcbd? Justify your answer. D 14. Underline the correct word to complete the sentence. To prove two triangles congruent, show that all adjacent / corresponding parts are congruent. A B C 15. Circle the name(s) for nacd. acute isosceles right scalene 16. Cross out the congruence statements that are NOT supported by the information in the figure. AD > CD BD > BD AB > CB /A > /C /ABD > /CBD /ADB > /CDB 17. You need congruence statements to prove two triangles congruent, so you can / cannot prove that nabd > ncbd. Theorem 4 1 Third Angles Theorem Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Use kabc and kdef above. 18. If m/a 5 74, then m/d If m/b 5 44, then m/e If m/c 5 62, then m/f 5. Problem 4 Proving Triangles Congruent Got It? Given: la O ld, AE O DC, EB O CB, BA O BD Prove: kaeb O kdcb If… /A > /D and /B > /E B A C E A D E F Then… /C > /F 21. You are given four pairs of congruent parts. Circle the additional information you need to prove the triangles congruent. A third pair A second pair A third pair of congruent of congruent of congruent sides angles angles B C D Copyright by Pearson Education, Inc. or its affiliates. All Rights Reserved. 6 Chapter 4 92

7 22. Complete the steps of the proof. 1) AE >, EB >, BA > 1) Given 2) /A > 2) Given 3) /ABE > 3) Vertical angles are congruent. 4) /E > 4) Third Angles Theorem 5) naeb > 5) Definition of > triangles Lesson Check Do you UNDERSTAND? If each angle in one triangle is congruent to its corresponding angle in another triangle, are the two triangles congruent? Explain. 23. Underline the correct word to complete the sentence. To disprove a conjecture, you need one / two / many counterexample(s). 24. An equilateral triangle has three congruent sides and three 608angles. Circle the equilateral triangles below. Copyright by Pearson Education, Inc. or its affiliates. All Rights Reserved. 25. Use your answers to Exercise 24 to answer the question. Math Success Check off the vocabulary words that you understand. congruent Need to review polygons Rate how well you can identify congruent polygons Now I get it! 93 Lesson 4-1 7

8 4-1 Think About a Plan Congruent Figures Algebra Find the values of the variables. Know 1. What do you know about the measure of each of the non-right angles? 2. What do you know about the length of each of the legs? 3. What types of triangles are shown in the figure? Need 4. What information do you need to know to find the value of x? 5. What information do you need to know to find the value of t? Plan 6. How can you find the value of x? What is its value? 7. How do you find the value of t? What is its value? Prentice Hall Geometry Teaching Resources 2 8

9 4-1 Practice Form G Congruent Figures Each pair of polygons is congruent. Find the measures of the numbered angles ΔCAT ΔJSD. List each of the following. 4. three pairs of congruent sides 5. three pairs of congruent angles WXYZ JKLM. List each of the following. 6. four pairs of congruent sides 7. four pairs of congruent angles For Exercises 8 and 9, can you conclude that the triangles are congruent? Justify your answers. 8. ΔGHJ and ΔIHJ 9. ΔQRS and ΔGHJ 10. Developing Proof Use the information given in the diagram. Give a reason that each statement is true. a. L Q b. LNM QNP c. M P d. LM QP, LN QN, MN PN e. ΔLNM ΔQNP Prentice Hall Gold Geometry Teaching Resources 3 9

10 4-1 Practice (continued) Form G Congruent Figures For Exercises 11 and 12, can you conclude that the figures are congruent? Justify your answers. 11. AEFD and EBCF 12. ΔFGH and ΔJKH Algebra Find the values of the variables Algebra ABCD FGHJ. Find the measures of the given angles or lengths of the given sides. 15. m B = 3y, m G = y CD = 2x + 3; HJ = 3x m C = 5z + 20, m H = 6z AD = 5b + 4; FJ = 3b LMNP QRST. Find the value of x. 20. Given: BD is the angle bisector of ABC. BD is the perpendicular bisector of AC. Prove: ΔADB ΔCDB 10 Prentice Hall Gold Geometry Teaching Resources 4

11 4-1 Practice Congruent Figures Form K Each pair of polygons is congruent. Find the measures of the numbered angles Use the diagram at the right for Exercises 3 7. ΔABC ΔXYZ. Complete the congruence statements. 3. AB To start, use the congruence statement to identify the points that correspond to A and B. A corresponds to B corresponds to 4. ZY 5. Z 6. BAC 7. B FOUR MANY. List each of the following. 8. four pairs of congruent angles 9. four pairs of congruent sides For Exercises 10 and 11, can you conclude that the figures are congruent? Justify your answers. 10. ΔSRT and ΔPRQ 11. ΔABC and ΔFGH Prentice Hall Foundations Geometry Teaching Resources 5 11

12 4-1 Practice (continued) Congruent Figures Form K 12. Given: AD and BE bisect each other. AB DE ; A D Prove: ΔACB ΔDCE Statements Reasons 1) AD and BE bisect each other. 1) Given AB DE, A D 2) AC CD, BC CE 2) 3) ACB DCE 3) 4) B E 4) 5) ΔACB ΔDCE 5) 13. If ΔACB ΔJKL, which of the following must be a correct congruence statement? A L AB JL B K ΔBAC ΔLKJ 14. Reasoning A student says she can use the information in the figure to prove ΔACB ΔACD. Is she correct? Explain. Algebra Find the values of the variables. 15. ΔXYZ ΔFED 16. ΔABD ΔCDB Algebra ΔFGH ΔQRS. Find the measures of the given angles or the lengths of the given sides. 17. m F = x + 24; m Q = 3x 18. GH = 3x 2; RS = x Prentice Hall Foundations Geometry Teaching Resources 6

13 4-1 Standardized Test Prep Congruent Figures Multiple Choice For Exercises 1 6, choose the correct letter. 1. The pair of polygons at the right is congruent. What is m/j? A B G F C H D J 2. The triangles at the right are congruent. Which of the following statements must be true? B D /A > /D AB > DE A E /B > /E BC > DF C F 3. Given the diagram at the right, which of the following must be true? F nxsf > nxtg nsxf > ngxt nfxs > nxgt nfxs > ngxt S X G 4. If nrst > nxyz, which of the following need not be true? /R > /X /T > /Z RT > XZ SR > YZ T 5. If nabc > ndef, m/a 5 50, and m/e 5 30, what is m/c? If ABCD > QRST, m/a 5 x 2 10, and m/q 5 2x 2 30, what is m/a? Short Response 7. Given: AB 6 DC, AD 6 BC, AB > CD, AD > CB Prove: nabd > ncdb A B D C Prentice Hall Geometry Teaching Resources 7 13

14 4-1 Solve It! You are working on a puzzle. You ve almost finished, except for a few pieces of the sky. Place the remaining pieces in the puzzle. How did you p figure out where to place the pieces? f They say you can t fit a square peg into a round hole. I wonder why that is. A 1 C B Lesson Quiz 1. If CDEF > KLMN, what are the congruent corresponding parts? 2. If nuvw > nefc, what is the measure of /FEC? U V C W F E 3. Do you UNDERSTAND? Suppose it is given that /C > /B, /D > /A, AE > BE, and CE > DE. Does that prove that the triangles are congruent? Justify your answer. D C E A B Answers Solve It! Lesson Quiz Piece 1 ﬁts in A, piece 2 in B, and piece 3 in C; explanations may vary. Sample: You can match up the parts that stick out with the parts that go in based on their size and location. 1. Sides: CD > KL, DE > LM, EF > MN, CF > KN; Angles: /C > /K, /D > /L, /E > /M, /F > /N No, the two triangles have congruent angles but not necessarily congruent sides. Prentice Hall Geometry Solve It/Lesson Quiz on Transparencies MatPac101071GeoResourceSampler.indd 14 7/1/10 10:51 AM

15 4-1 Additional Problems Congruent Figures Problem 1 If RSTU > WXYZ, what are the congruent corresponding parts? Problem 2 The sides of a roof suggest congruent triangles. What is m/1? A. 90 B. 48 C. 42 D Prentice Hall Geometry Additional Problems 43 15

16 4-1 Additional Problems (continued) Congruent Figures Problem 3 Are the triangles congruent? Justify your answer. A B C D Problem 4 Given: RS > RU, TS > TU, /S > /U, /SRT > /URT S T U Prove: nrst > nrut R Prentice Hall Geometry Additional Problems 44 16

17 4-1 Reteaching Congruent Figures Given ABCD QRST, find corresponding parts using the names. Order matters. For example, This shows that A corresponds to Q. Therefore, A Q. For example, This shows that BC corresponds to RS. Exercises Therefore, BC RS. Find corresponding parts using the order of the letters in the names. 1. Identify the remaining three pairs of corresponding angles and sides between ABCD and QRST using the circle technique shown above. Angles: ABCD ABCD ABCD Sides: ABCD ABCD ABCD QRST QRST QRST QRST QRST QRST 2. Which pair of corresponding sides is hardest to identify using this technique? Find corresponding parts by redrawing figures. 3. The two congruent figures below at the left have been redrawn at the right. Why are the corresponding parts easier to identify in the drawing at the right? 4. Redraw the congruent polygons at the right in the same orientation. Identify all pairs of corresponding sides and angles. 5. MNOP QRST. Identify all pairs of congruent sides and angles. Prentice Hall Geometry Teaching Resources 9 17

19 4-1 ELL Support Congruent Figures Concept List algebraic equation congruent angles congruent triangles angle measure congruent polygons proof congruency statement congruent segments segment measure Choose the concept from the list above that best represents the item in each box. 1. GH ST 2. m A = YZ = MN 5. ΔABC ΔXYZ 6. Given: BD is the angle bisector of ABC, and BD is the perpendicular bisector of AC. Prove: ΔADB ΔCDB 7. m H = 5x m W = x + 28 Solve 5x = x + 28 to find the measures of H and W. 8. BC = 3 cm 9. ADB and SDT are vertical angles. So, ADB SDT. Prentice Hall Geometry Teaching Resources 1 19

20 Activity: Create Your Own Logo 4-1 Congruent Figures Materials Graph paper Colored pencils or crayons A logo is an identifying statement often represented in symbolic form. With exposure from advertising, many corporate logos have become familiar. Work in a group to identify corporation logos that use these shapes. 1. triangles 2. circles 3. squares Logos often include congruent figures to help establish symmetric eye-catching forms. Identify the congruent figure in each logo Design a logo of your own, using at least two sets of congruent triangles. Other congruent figures also may be used. Use graph paper, and include color in your design. Prentice Hall Geometry Activities, Games, and Puzzles 26 20

21 Name Class Date 4-3 Game: Big Hitters Triangle Congruence by ASA and AAS Setup Your teacher will divide the class into teams of 5 students. Cut out the set of diagrams below. As a team, sit in a circle and place the diagrams in the center, face down. Game Play Certain theorems, properties, and definitions are used more frequently than others to find congruent parts when proving that two triangles are congruent. You might call them big hitters. Being able to recognize when these big hitters may be used is a big advantage when writing proofs. As a team, look for ways to apply the big hitters. In each round, a different student is to reveal a diagram. Work as a team to write down as many big hitters as you can that could likely apply to the diagram. When your teacher calls time, he or she will reveal the correct answers and your team will earn a point for each correctly identified big hitter. A point is subtracted for incorrect answers. After 9 rounds, the team with the greatest number of points wins. Prentice Hall Geometry Activities, Games, and Puzzles 29 21

22 Name Class Date Puzzle: Cage the Monster 4-4 Using Corresponding Parts of Congruent Triangles A proof with multiple pairs of congruent triangles can seem like a monster. But, you can control the monster if you can master the diagram. Build a fence around each monster by stating the shared congruent parts for the given pairs of congruent triangles. The first problem has been started for you. Prentice Hall Geometry Activities, Games, and Puzzles 30 22

23 4-1 Enrichment Congruent Figures Shared Implications Sometimes different statements share one or more implications. For example, QR ST and QR is the perpendicular bisector of ST share the implication that QR meets ST at a right angle. The statements below refer to the diagram at the right. 1. DJ JK ; 2. DJ AD ; 3. AD JK ; 4. A K; 5. DX JX ; 6. AD KJ ; 7. AK bisects DJ ; 8. DJ bisects AK ; 9. m D = m J = 90 Identify shared implications and reduce the number of given statements. 1. What implication is shared by Statement 5 and Statement 7? 2. What implication is shared by Statement 3 and Statement 4? 3. Which two statements share at least one implication with Statement 9? 4. Can you prove ΔADX ΔKJX using only five of the statements above? If so, identify them, then complete the proof. 5. Can you prove ΔADX ΔKJX using only four of the statements above? If so, identify them, then complete the proof. 6. Can you prove ΔADX ΔKJX using only three of the statements above if the only way to prove triangles congruent is through the definition of congruent triangles? If so, identify them, then complete the proof. Prentice Hall Geometry Teaching Resources 8 23

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28 Chapter 4 Quiz 1 Lessons 4-1 through 4-3 Form G Do you know HOW? 1. Two triangles have the following pairs of congruent sides: BD FJ, DG JM, and GB MF. Write the congruence statement for the two triangles. QRS TUV. Name the angle or side that corresponds to the given part. 2. Q 3. RS 4. S 5. QS State the postulate or theorem that can be used to prove the triangles congruent. If you cannot prove the triangles congruent, write not enough information. Use the diagram below. Tell why each statement is true. 10. A C 11. AXB CXD 12. ABX CDX Do you UNDERSTAND? 13. Given: LM NO; LMO NOM Prove: LMO NOM 14. Reasoning Explain why it is not possible to have a Side-Side-Angle congruence postulate or theorem. Draw a picture if necessary. 28 Prentice Hall Gold Geometry Teaching Resources 71

29 Chapter 4 Quiz 2 Form G Lessons 4-4 through 4-7 Do you know HOW? Explain how to use congruent triangles to prove each statement true Find the values of x and y Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL Do you UNDERSTAND? 7. Reasoning Complete the proof by filling in the missing statements and reasons. Given: 1) 2) 3) 4) 5) Prove: Statements 1) Reasons 2) Reflexive Property of Congruence 3) 4) 5) Segment Addition Postulate Prentice Hall Gold Geometry Teaching Resources 72 29

30 Chapter 4 Test Form G Do you know HOW? State the postulate or theorem you would use to prove each pair of triangles congruent. If the triangles cannot be proven congruent, write not enough information Find the value of x and y CGI MPR. Name all of the pairs of corresponding congruent parts. Prentice Hall Gold Geometry Teaching Resources 73 30

31 Chapter 4 Test (continued) Form G Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. 15. Given: LM LK; LN LJ 16. Given: ABC DCB; DBC ACB 17. Given: E D DCF EFC 18. Given: HI JG Do you UNDERSTAND? 19. Reasoning Complete the following proof by providing the reason for each statement. Given: 1 2; WX Prove: 3 4 Statements 1) 1 2; WX ZY 2) WP ZP 3) WXP ZYP 4) XP YP 5) 3 4 ZY 20. Reasoning Write a proof for the following: Given: BD 1) 2) 3) 4) 5) Reasons AC, D is the midpoint of AC. Prove: BC BA Prentice Hall Gold Geometry Teaching Resources 74 31

32 Chapter 4 Find the Errors! For use with Lessons 4-1 through 4-2 For each exercise, identify the error(s) in planning the solution or solving the problem. Then write the correct solution. 1. If kabc O kgkq, what are the congruent corresponding parts? B K A C Q G Sides: AC > QG; AB > QK ; BC > KG Angles: /A > /Q; /B > /K ; /C > /G 2. Given: PO n MN, PO O MN Prove: kmpn O konp P O M N Statements 1) PO 6 MN 2) PO > MN 3) PN > PN 4) nmpn > nonp Reasons 1) Given 2) Given 3) Reflexive Property of > 4) SS Postulate 3. What other information do you need to prove the triangles congruent by SAS? Explain. B E A C D F None. The triangles have two pairs of congruent sides (AB > DE, and BC > EF ) and one pair of congruent angles (/BAC > /EFD). So, the triangles are congruent by SAS. Prentice Hall Geometry Find the Errors! 19 32

33 Chapter 4 Find the Errors! For use with Lessons 4-3 through 4-5 For each exercise, identify the error(s) in planning the solution or solving the problem. Then write the correct solution. 1. Which two triangles are congruent by ASA? Explain. nabc > ndef are congruent by ASA. They each have two pairs of congruent angles and one pair of congruent sides. A B C D E F H I G 2. Given: la O lc BD bisects ladc Prove: kadb O kcdb D A B C Statements 1) /A > /C 2) BD bisects /ADC 3) BD > BD 4) nadb > ncdb Reasons 1) Given 2) Given 3) Reflexive Property of > 4) AAS Theorem 3. Given: AB O CD AD O BC Prove: la O lc B C Statements Reasons A D 1) AB > CD 1) Given 2) AD > BC 2) Given 3) BD > BD 3) Reflexive Property of > 4) /A > /C 4) Corresponding parts of > triangles are >. 4. What are the values of x and y? y x 5 808, y x Prentice Hall Geometry Find the Errors! 21 33

34 Chapter 4 Find the Errors! For use with Lessons 4-6 through 4-7 For each exercise, identify the error(s) in planning the solution or solving the problem. Then write the correct solution. 1. On the diagram shown, ln and lq are right angles and NP O MQ. Are knpm and kqmp congruent? Write a paragraph proof. /N and /Q are right angles. So, nnpm and nqmp are right triangles. Also, NP > MQ. Therefore, nnpm > nqmp by the Hypotenuse Leg Theorem. N M P Q D 2. Given: DC ‘ AE, DE O AC B is the midpoint of AE Prove: kbde O kbca A B E Statements 1) DC ‘ AE 2) DE > AC 3) B is the midpoint of AE 4) AB > BE 5) nbde and nbca are right ns 6) nbde > nbca Reasons 1) Given 2) Given 3) Given 4) Definition of midpoint 5) Definition of right triangle 6) Hypotenuse Leg Theorem C 3. In the diagram, kade O kdab. What is their common side or angle? A B /C C D E Prentice Hall Geometry Find the Errors! 23 34

35 Performance Tasks Chapter 4 Task 1 Draw and label three pairs of triangles to illustrate the Side-Side-Side, Angle-Side-Angle, and Side-Angle-Side Postulates. One pair of triangles should share a common side. The figures should provide enough information to prove that they are congruent. Write the congruence statements for each pair. Task 2 S A rhombus is a quadrilateral with four congruent sides. Given: RSTQ is a quadrilateral, /SRT > /STR > /RTQ > /TRQ. Prove: RSTQ is a rhombus. R T Q Prentice Hall Geometry Teaching Resources 79 35

36 Performance Tasks (continued) Chapter 4 Task 3 You need to design a company logo. The requirements for the logo are as listed: The logo must include at least six triangles. Some of the triangles should overlap. Some of the triangles should share sides. One triangle needs to be isosceles. One triangle needs to be equilateral. At least two pairs of triangles should be congruent pairs. Use a straightedge, compass, and protractor to aid in your design. Label the vertices of the triangles and describe as many congruencies as you can (sides and angles). Describe two pairs of congruent triangles in your design and justify how you know they are congruent. Include references to geometric theorems and postulates. Prentice Hall Geometry Teaching Resources 80 36

37 Extra Practice Chapter 4 Lesson 4-1 SAT GRE. Complete each congruence statement. 1. S 2. GR 3. E 4. AT 5. ΔERG 6. EG 7. ΔREG 8. R State whether the figures are congruent. Justify your answers. 9. ABF; EDC 10. TUV; UVW 11. XYZV; UTZV 12. ABD; EDB Lessons 4-2 and 4-3 Can you prove the two triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not possible and tell what other information you would need Prentice Hall Geometry Extra Practice 12 37

38 Extra Practice (continued) Chapter Given: PX PY, ZP bisects XY. 18. Given: 1 2, 3 4, PD PC, Prove: PXZ PYZ P is the midpoint of AB. Prove: ADP BCP 19. Given: 1 2, 3 4, AP DP 20. Given: MP NS, RS PQ, MR Prove: ABP > DCP Prove: MQP NRS NQ Lesson Given: LO MN, LO MN Prove: MLN ONL Prove: TO ES 22. Given: OTS OES, EOS OST 23. Given: 1 2, 3 4, 24. Given: PO = QO, 1 2, M is the midpoint of PR Prove: PMQ RMQ Prove: A B Prentice Hall Geometry Extra Practice 13 38

39 Extra Practice (continued) Chapter 4 Lesson 4-5 Find the value of each variable Given: 5 6,PX PY 29. Given: AP BP,PC PD Prove: PAB is isosceles. Prove: QCD is isosceles. Lessons 4-6 and 4-7 Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL Given: M is the midpoint of AB, MC AC,MD BD, 1 2 Prove: ACM BDM 35. The longest leg of ABC, AC, measures 10 centimeters. BC measures 8 centimeters. You measure two of the legs of XYZ and find that AC XZ and BC YZ. Can you conclude that two triangles to be congruent by the HL Theorem? Explain why or why not. Prentice Hall Geometry Extra Practice 14 39

40 Chapter 4 Project: Tri, Tri Again Beginning the Chapter Project Have you ever wondered how bridges stay up? How do such frail-looking frameworks stretch through the air without falling? How can they withstand the twisting forces of hurricane winds and the rumbling weight of trucks and trains? Part of the answer lies in the natural strength of triangles. In your project for this chapter, you will explore how engineers use triangles to construct safe, strong, stable structures. You then will have a chance to apply these ideas as you design and build your own bridge with toothpicks or craft sticks. You will see how a simple shape often can be the strongest one. Activities Activity 1: Modeling Many structures have straight beams that meet at joints. You can use models to explore ways to strengthen joints. Cut seven cardboard strips approximately 6 in. by 1 2 in. Make a square frame and a triangular frame. Staple across the joints as shown. With your fingertips, hold each model flat on a desk or table, and try to Change its shape. Which shape is more stable? Cut another cardboard strip, and use it to form a brace for the square frame. Is it more rigid? Why does the brace work? Activity 2: Observing Visit local bridges, towers, or other structures that have exposed frameworks. Examine these structures for ideas you can use when you design and build a bridge later in this project. Record your ideas. Sketch or take pictures of the structures. On the sketches or photos, show where triangles are used for stability. 40 Prentice Hall Geometry Teaching Resources 84

42 Chapter 4 Project Manager: Tri, Tri Again Getting Started As you work on the project, you will need a sheet of cardboard, a stapler, 100 toothpicks or 30 craft sticks, and glue. Keep this Project Manager and all your work for the project in a folder or an envelope. Checklist Activity 1: cardboard frames Activity 2: observing bridges Activity 3: three-dimensional models toothpick bridge Suggestions Push or pull the models only along the plane of the frame. Look for small design features that are used repeatedly. Use glue that is strong but quick-drying. Test small parts of the bridge before building the entire structure. Also, decide in advance in what order you will assemble and glue the different sections. Scoring Rubric 4 The toothpick bridge meets all specifications. The diagrams and explanations are clear. Geometric language was used appropriately and correctly. A complete account of the experiments was given, including how they led to improved designs. 3 The toothpick bridge meets or comes close to meeting all specifications. The diagrams and explanations are understandable but may contain a few minor errors. Most of the geometric language is used appropriately and correctly. Evidence was shown of at least one experimental model prior to the finished model. 2 The toothpick bridge does not meet specifications. Diagrams and explanations are misleading or hard to follow. Geometric terms are completely lacking, used sparsely, or often misused. The model shows little effort and no evidence of testing of preliminary designs. 1 Major elements of the project are incomplete or missing. 0 Project is not handed in or shows no effort. Your Evaluation of Project Evaluate your work, based on the Scoring Rubric. Teacher s Evaluation of Project 42 Prentice Hall Geometry Teaching Resources 86

43 T E A C H E R I N S T R U C T I O N S Chapter 4 Project Teacher Notes: Tri, Tri Again About the Project Students will explore how engineers use triangles to construct safe, strong, stable structures. Then they will apply these ideas to build their own bridges, using toothpicks or craft sticks. Introducing the Project Ask students whether they have ever built towers using playing cards. Ask them how they placed the first cards and why. Have students make towers using playing cards. Activity 1: Modeling Students will discover that triangles are more stable or rigid than quadrilaterals. Discuss with students real-world examples in which triangles are used for stability, such as ironing boards, scaffolding, and frames of roofs. Activity 2: Observing If students cannot find any local structures with exposed frameworks, suggest that they look in books or on the Internet for pictures of architecture or construction. Activity 3: Investigating Have students work in groups, keeping a log of the different models they make in their attempt to find one that supports the weight of the geometry book. Have groups compare the successful models and discuss their similarities and differences. Finishing the Project You may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their products. Ask students to share how they selected their final bridge design. Ask students to submit their best models for a bridge-breaking competition, an event to which you could invite parents and the community. Prentice Hall Geometry Teaching Resources 83 43

44 Cumulative Review Chapters 1 4 Multiple Choice Use the diagram for Exercises 1 and 2. Line < is parallel to line m. 1. Which best describes /1 and /5? alternate interior angles alternate exterior angles corresponding angles , m same-side exterior angles 2. Which best describes /6 and /7? vertical angles corresponding angles alternate exterior angles linear pair 3. If an animal is a mammal, then it has fur. What is the conclusion of this conditional? An animal is a mammal. Mammals have fur. The animal has fur. Not all animals have fur. 4. Two of what geometric figure are joined at a vertex to form an angle? points planes rays lines 5. If WZ 5 80, what is the value of y? 2y 2y 3 3y 7 W X Y Z 6. If nabc > ndef, which is a correct congruence statement? /B > /D AB > EF CA > FD /A > /C 7. Which can be used to justify stating that nfgh > njkl? G K ASA SSS SAS AAS F H J L 8. Which postulate can be used to justify stating that nlmn > npqr? L P ASA SSS SAS AAS M N Q R 44 Prentice Hall Geometry Teaching Resources 81

45 Cumulative Review (continued) Chapters 1 4 Short Response 9. What is the midpoint of a segment with endpoints at (22, 2) and (5, 10)? Use the figure at the right for Exercises Given: AB > ED, BC > DC 10. Which reason could you use to prove AC > EC? 11. Which reason could you use to prove /C > /C? A B F C D E 12. Which reason could you use to prove nacd > necb? 13. What is the slope of a line that passes through (23, 5) and (4, 3)? 14. What is the slope of a line that is perpendicular to the line that passes through (22, 22) and (1, 3)? Extended Response 15. Draw nabc > nefg. Write all six congruence statements. 16. The coordinates of rectangle HIJK are H(24, 1), I(1, 1), J(1, 22), and K(24, 22). The coordinates of rectangle LMNO are L(21, 3), M(2, 3), N(2, 23), and O(21, 23). Are these two rectangles congruent? Explain. If not, how could you change the coordinates of one of the rectangles to make them congruent? Prentice Hall Geometry Teaching Resources 82 45

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