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3 APPLY ASSIGNMENT GUIDE BASIC Day 1: pp. 216–219 Exs. 6–21 Day 2: pp. 217–219 Exs. 22–28, 32–37, 39–46 AVERAGE Day 1: pp. 216–219 Exs. 6–21 Day 2: pp. 217–219 Exs. 22–28, 32–37, 39–46 GUIDED PRACTICE Vocabulary Check ✓ Concept Check ✓ 2. The congruent angles are not the angles included between the congruent sides. Skill Check ADVANCED ✓ Day 1: pp. 216–219 Exs. 6–21 Day 2: pp. 217–219 Exs. 22–28, 30–46 1. Sketch a triangle and label its vertices. Name two sides and the included angle between the sides. See margin. 2. ERROR ANALYSIS Henry believes he can use the information given in the diagram and the SAS Congruence Postulate to prove the two triangles are congruent. Explain Henry’s mistake. LOGICAL REASONING Decide whether enough information is given to prove that the triangles are congruent. If there is enough information, tell which congruence postulate you would use. 3. ¤ABC, ¤DEC B BLOCK SCHEDULE E H D yes; SAS Congruence Postulate K yes; SSS Congruence Postulate PRACTICE AND APPLICATIONS Extra Practice to help you master skills is on p. 809. 12. no 13. yes; SAS Congruence Postulate 14. yes; SSS Congruence Postulate 15. yes; SAS Congruence Postulate 16. no 17. yes; SSS Congruence Postulate STUDENT HELP S R J no STUDENT HELP q P G NAMING SIDES AND INCLUDED ANGLES Use the diagram. Name the included angle between the pair of sides given. Æ Æ Æ Æ Æ Æ 6. JK and KL ™JKL Æ Æ Æ Æ 7. PK and LK 8. LP and LK ™KLP HOMEWORK CHECK To quickly check student understanding of key concepts, go over the following exercises: Exs. 8, 16, 18, 20, 26, 28, 32, 34. See also the Daily Homework Quiz: • Blackline Master (Chapter 4 Resource Book, p. 55) • Transparency (p. 28) 5. ¤PQR, ¤SRQ F C pp. 216–219 Exs. 6–21 (with 4.2) pp. 216–219 Exs. 22–28, 30–37, 39–46 (with 4.4) EXERCISE LEVELS Level A: Easier 6–17 Level B: More Difficult 18–37 Level C: Most Difficult 38 4. ¤FGH, ¤JKH A 9. JL and JK Æ 10. KL and JL ™JLK Æ 11. KP and PL J ™LKP L ™KJL ™KPL K P LOGICAL REASONING Decide whether enough information is given to prove that the triangles are congruent. If there is enough information, state the congruence postulate you would use. 12–17. See margin. 12. ¤UVT, ¤WVT 13. ¤LMN, ¤TNM L T 14. ¤YZW, ¤YXW T Z W Y W V M U N X HOMEWORK HELP Example 1: Exs. 18, 20–28 Example 2: Exs. 19–28 Example 3: Exs. 12–17 Example 4: Exs. 20–28 Example 5: Exs. 30, 31 Example 6: Exs. 33–35 15. ¤ACB, ¤ECD A 216 T 17. ¤GJH, ¤HLK U G J L C B 216 16. ¤RST, ¤WVU D Chapter 4 Congruent Triangles E R S V W H M K DEVELOPING PROOF In Exercises 18 and 19, use the photo Æ Æ Æ Æ of the Navajo rug. Assume that BC £ DE and AC £ CE . COMMON ERROR EXERCISE 2 Students may incorrectly say that two triangles can be proven congruent using the SAS Congruence Postulate when they are given two pairs of corresponding congruent sides and the corresponding nonincluded angles. Demonstrate that SSA is not a congruence postulate. 18. What other piece of information is needed to prove that ¤ABC £ ¤CDE using the Æ Æ SSS Congruence Postulate? AB £ CD 19. What other piece of information is needed to prove that ¤ABC £ ¤CDE using the SAS Congruence Postulate? ™ACB £ ™CED 20. DEVELOPING PROOF Complete the proof by supplying the reasons. Æ Æ Æ Æ GIVEN 䉴 EF £ GH , H E G F STUDENT HELP NOTES Homework Help Students can find help for Exs. 23–24 at www.mcdougallittell.com. The information can be printed out for students who don’t have access to the Internet. FG £ HE PROVE 䉴 ¤EFG £ ¤GHE Statements 23. It is given that Æ Æ ˘ SP £TP and that PQ bisects ™SPT. Then, by the definition of angle bisector, ™SPQ £ ™TPQ. Æ Æ PQ £ PQ by the Reflexive Property of Congruence, so ¤SPQ £ ¤TPQ by the SAS Congruence Postulate. 24. It is given that Æ Æ £ RT and PT Æ Æ QT £ ST . ™PTQ £ ™RTS by the Vertical Angles Theorem. Then ¤PQT £ ¤RST by the SAS Congruence Postulate. INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with paragraph proofs. Æ Æ Æ Æ Æ Æ 1. Sample answer: Reasons ?㛭㛭㛭 Given 1. 㛭㛭㛭㛭㛭 ?㛭㛭㛭 Given 2. 㛭㛭㛭㛭㛭 Property ?㛭㛭㛭 Reflexive 3. 㛭㛭㛭㛭㛭 of Congruence ?㛭㛭㛭 SSS Congruence Postulate 4. 㛭㛭㛭㛭㛭 1. EF £ GH 2. FG £ HE 3. GE £ GE 4. ¤EFG £ ¤GHE TWO-COLUMN PROOF Write a two-column proof. 21, 22.See margin. Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ 21. GIVEN 䉴 NP £ QN £ RS £ TR, 22. GIVEN 䉴 AB £ CD, AB ∞ CD PQ £ ST PROVE 䉴 ¤ABC £ ¤CDA PROVE 䉴 ¤NPQ £ ¤RST N S A T D 1 Y X Z åX is included between X苶Y苶 and X苶Z苶, åY is included between X苶Y苶 and Y苶Z苶, and åZ is included between X苶Z苶 and Y苶Z苶. 21. Statements (Reasons) 1. N 苶P苶 • Q 苶N 苶 • R苶S苶 • T苶R苶, P苶Q 苶 • S苶T苶 (Given) 2. †NPQ • †RST (SSS Congruence Postulate) 22. See Additional Answers beginning on page AA1. 2 q P R B C PARAGRAPH PROOF Write a paragraph proof. 23, 24. See margin. Æ ˘ 23. GIVEN 䉴 PQ bisects ™SPT, Æ Æ SP £ TP PROVE 䉴 ¤SPQ £ ¤TPQ Æ Æ Æ PROVE 䉴 ¤PQT £ ¤RST q P P S T T q Æ 24. GIVEN 䉴 PT £ RT, QT £ ST S R 4.3 Proving Triangles are Congruent: SSS and SAS 217 217 Software Help Instructions for several software packages are available in blackline format in the Chapter 4 Resource Book, p. 44 and at www.mcdougallittell.com. 25. Statements (Reasons) 1. A 苶C苶 • B 苶C苶, M is the midpoint of A 苶B 苶. (Given) 2. A 苶M 苶• B 苶M 苶 (Definition of midpoint) 3. C苶M 苶 • C苶M 苶 (Reflexive Property of Congruence) 4. †ACM • †BCM (SSS Congruence Postulate) 26. Statements (Reasons) 1. B 苶C苶 • A 苶E苶, B 苶D 苶• A 苶D 苶, D 苶E苶 • D 苶C苶 (Given) 2. BD = AD, DE = DC (Definition of congruent segments) 3. BD + DE = AD + DC (Addition property of equality) 4. BD + DE = BE, AD + DC = AC (Segment Addition Postulate) 5. BE = AC (Substitution property) 6. B 苶E苶 • A 苶C苶 (Definition of congruent segments) 7. A 苶B 苶• A 苶B 苶 (Reflexive Property of Congruence) 8. †ABC • †BAE (SSS Congruence Postulate) 33–35, 38, 46. See Additional Answers beginning on page AA1. 27. Since it is given that Æ Æ Æ PA £ PB £ PC and Æ Æ AB £ BC , ¤PAB £ ¤PBC by the SSS Congruence Postulate. For Lesson 4.3: • Practice Levels A, B, and C (Chapter 4 Resource Book, p. 45) • Reteaching with Practice (Chapter 4 Resource Book, p. 48) • See Lesson 4.3 of the Personal Student Tutor For more Mixed Review: • Search the Test and Practice Generator for key words or specific lessons. 218 Æ STUDENT HELP NE ER T Æ Æ Æ Æ Æ 26. GIVEN BC £ AE, BD £ AD, Æ M is the midpoint of AB. PROVE ¤ACM £ ¤BCM Æ DE £ DC PROVE ¤ABC £ ¤BAE C B C D A M B Æ Æ Æ Æ E A Æ 27. GIVEN PA £ PB £ PC, Æ Æ Æ Æ Æ Æ 28. GIVEN CR £ CS, QC fi CR, AB £ BC QC fi CS PROVE ¤PAB £ ¤PBC PROVE ¤QCR £ ¤QCS q P C C R B A S TECHNOLOGY Use geometry software to draw a triangle. Draw a line 29. and reflect the triangle across the line. Measure the sides and the angles of the new triangle and tell whether it is congruent to the original one. See margin. SOFTWARE HELP Visit our Web site www.mcdougallittell.com to see instructions for several software applications. Æ 25. GIVEN AC £ BC, 29. The new triangle and the original triangle are congruent. Writing Explain how triangles are used in the object shown to make it more stable. 30. 31. 30, 31. Sample answers are given. 30. The cross pieces form triangles which are rigid, ensuring that the supports keep their shape. 31. The struts that go from the body of the plane to the wing form triangles, making the wing structure rigid. ADDITIONAL PRACTICE AND RETEACHING PROOF Write a two-column proof or a paragraph proof. 25–28. See margin. Æ 28. It is given that CR £ Æ Æ CS and that QC is perpendicular to both Æ Æ CR and CS . If two lines are perpendicular, then they intersect to form four right angles, so ™QCR and ™QCS are right angles. By the Right Angle Congruence Theorem, ™QCR £ ™QCS. By the Reflexive Property Æ of Congruence, QC £ Æ QC . Then ¤QCR £ ¤QCS by the SAS Congruence Postulate. INT STUDENT HELP NOTES 32. Use the method described in the activity on page 213. Note, however, that the same compass setting will be used to construct the legs of the triangle. 32. CONSTRUCTION Draw an isosceles triangle with vertices A, B, and C. Use a compass and straightedge to construct ¤DEF so that ¤DEF £ ¤ABC. See margin. xy USING ALGEBRA Use the Distance Formula and the SSS Congruence Postulate to show that ¤ABC £ ¤DEF. 33–35. See margin. 33. 34. y C E D y y E D D C A E 2 1 A B F 5 x C Chapter 4 Congruent Triangles F 1 B 1 218 35. F x A B 1 x Test Preparation Æ Æ Æ Æ 36. MULTIPLE CHOICE In ¤RST and ¤ABC, RS £ AB, ST £ BC, and Æ Æ TR £ CA. Which angle is congruent to ™T? C A ¡ B ¡ ™R C ¡ ™A D ¡ ™C 4 ASSESS cannot be determined 37. MULTIPLE CHOICE In equilateral ¤DEF, a segment is drawn from point F Æ Transparency Available Use the diagram. Name the included angle between the pairs of sides given. to G, the midpoint of DE. Which of the statements below is not true? B ★ Challenge A ¡ Æ B ¡ Æ DF £ EF Æ Æ DG £ DF C ¡ Æ Æ DG £ EG D ¡ 38. CHOOSING A METHOD Describe how EXTRA CHALLENGE www.mcdougallittell.com ¤DFG £ ¤EFG y to show that ¤PMO £ ¤PMN using the SSS Congruence Postulate. Then find a way to show that the triangles are congruent using the SAS Congruence Postulate. You may not use a protractor to measure any angles. Compare the two methods. Which do you prefer? Why? See margin. N D F 1 O 1 x P SCIENCE CONNECTION Find an important angle in the photo. Copy the angle, extend its sides, and use a protractor to measure it to the nearest degree. (Review 1.4) 39. 40. G H 1. D 苶G 苶 and G 苶E苶 åDGE 2. E苶H 苶 and G 苶H 苶 åEHG Using the diagram above, decide whether enough information is given to prove that the triangles are congruent. If there is enough information, state the congruence postulate you would use. 3. 䉭DGF, 䉭EGH no 4. 䉭DGE, 䉭EGH yes; SSS Congruence Postulate EXTRA CHALLENGE NOTE Challenge problems for Lesson 4.3 are available in blackline format in the Chapter 4 Resource Book, p. 52 and at www.mcdougallittell.com. USING PARALLEL LINES Find m™1 and m™2. Explain your reasoning. (Review 3.3 for 4.4) 41. 42. 43. 1 1 129ⴗ 2 1 ADDITIONAL TEST PREPARATION 1. WRITING Explain the difference between the SSS Congruence Postulate and the SAS Congruence Postulate. 2 2 57ⴗ LINE RELATIONSHIPS Find the slope of each line. Identify any parallel or perpendicular lines. (Review 3.7) 44–46. See margin. 44. 45. y 3 ¯˘ 3 slope of BD = }}; 5 ¯˘ ¯˘ AC fi BD ¯˘ 45. slope of EF = º2, slope of ¯˘ ¯˘ ¯˘ GH = º2, EF ∞ GH E M MIXED REVIEW 39, 40. Sample answers are given. 39. The measure of each of the angles formed by two adjacent “spokes” is about 60°. 40. The measure of each of the angles formed by two adjacent sides of a cell of the honeycomb is about 120°. 41. m™2 = 57° (Vertical Angles Theorem), m™1 = 180° º m™2 = 123˚ (Consecutive Interior Angles Theorem) 42. m™1 = 180° º 129° = 51° (Linear Pair Postulate), m™2 = m™1 = 51° (Alternate Exterior Angles Theorem) 43. m™1 = 90° (Corresponding Angles Postulate), m™2 = 90° (Alternate Interior Angles Theorem or Vertical Angles Theorem) ¯˘ 5 44. slope of AC = º }}, DAILY HOMEWORK QUIZ A E D 1 x C y G 1 1 2 B 46. y F œ x H R P 1 1 4.3 Proving Triangles are Congruent: SSS and SAS x Sample answer: Both are used to prove triangles congruent. SSS is used when three sides of one triangle are congruent to three sides of a second triangle. SAS is used when two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle. 219 219