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CHAPTER 5 Chapter Opener 20. B; this statement matches the corresponding vertices in order. Chapter Readiness Quiz (p. 232) 1. D 2. G 3. D 21. a R a N 22. LN **** PR **** 23. a RPQ a NLM 24. QP **** ML **** 25. T MNL T QRP Lesson 5.1 5.1 Checkpoint (pp. 234–235) 1. Corresponding Angles Corresponding Sides aS aY ST **** YX **** aU aZ SU **** YZ **** aT aX UT ZX **** **** 28. no 29. yes 30. yes 31. no 32. no 33. Corresponding Angles 2. Because DF **** **** AC , you know that DF AC 3. 4. Yes; the congruent sides are marked on the diagram, so XY ZW , XV **** **& **** ZV **** , and YV **** WV **&. Vertical angles are congruent. aX aZ Alternate Interior Angles Theorem aY aW Alternate Interior Angles Theorem aP aS QP **** TS **** aQ aT QR **** TU **** aR aU PR SU **** **** 34. Corresponding Angles in order. a XVY a ZVW AC EF **** **** aC aF AB ED **** **** aB aD CB **** FD **** Since all corresponding parts are congruent, T ABC T EDF. 35. 5.1 Guided Practice (p. 236) A 1. corresponding angles 2. neither 4. neither 5. a F 6. a Z 7. ED **** 8. YZ **** 9. ma P ma L 105 F B E A C ma B ma E 100 12. LN PR 8.5 13. Yes; the congruent sides are marked on the diagram, so FE **** LK ****, FG **** **** LJ , and GE **** JK ****. Two pairs of congruent angles are also marked on the diagram, so a E a K and a G a J. The third pair of corresponding angles are right angles and are congruent since all right angles are congruent. All corresponding parts are congruent, so T EFG T KLJ. 5.1 Practice and Applications (pp. 236–239) 14. corresponding angles 15. neither 16. neither 17. corresponding sides 18. corresponding sides 19. corresponding angles D B 37. DF AC 5; 11. QR MN 11.6 Copyright © McDougal Littell Inc. All rights reserved. 36. C F 10. ma M ma Q 45 Corresponding Sides aA aE Since all corresponding parts are congruent, T XYV T ZWV. 3. corresponding sides Corresponding Sides Since all corresponding parts are congruent, T QPR T TSU. Because a B a E, you know that ma B ma E 28. 3. B; This statement matches up the corresponding vertices 26. T PRQ T LNM 27. yes 39. AB DE 6; ma F ma C 50 E D 38. AB DE 14; ma F ma C 39 40. BC EF 5; ma D ma A 27 41. C; this statement does not match up the corresponding vertices in order. 42. Corresponding Angles Corresponding Sides aA aD AB DF **** **** aB aF AC DE **** **** aC aE CB **** EF **** Since all corresponding parts are congruent, T ABC T DFE. Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 69 Chapter 5 continued 43. Corresponding Angles Corresponding Sides aJ aP JK **** PN **** aK aN JL PM **** **& aL aM KL NM **** **& Since all corresponding parts are congruent, T JKL T PNM. 58. obtuse; 62 52 22 59. acute; 102 52 92 60. obtuse; 142 52 122 5.1 Algebra Skills (p. 239) 61. 11.95 62. 3.72 63. 19.33 64. 0.74 65. 17.076 66. 3.222 44. The triangles are not congruent. Lesson 5.2 45. Corresponding Angles a BCA a DCE Vertical Angles Theorem aB aD Alternate Interior Angles Theorem aA aE Alternate Interior Angles Theorem 5.2 Activity (p. 240) 1. All triangles made with two pencils and a 45 angle appear to be congruent. 2. You need to show that the angle formed by the two Corresponding Sides known sides of one triangle is congruent to the angle formed by the two known sides of the other triangle. BC **** DC **** AC EC **** **** 5.2 Checkpoint (p. 244) AB ED **** **** Since all corresponding parts are congruent, 1. Statements Reasons T ABC T EDC. 1. CB **** CE **** 1. Given 46. Corresponding Angles 2. **** AC DC **** 2. Given Corresponding Sides a LJM a JLK JM **** LK **** 3. a BCA a ECD 3. Vertical Angles Theorem a JLM a LJK ML **** KJ **** 4. T BCA T ECD 4. SAS Congruence Postulate aM aK JL **** **** LJ 5.2 Guided Practice (p. 245) Since all corresponding parts are congruent, T JML T LKJ. 1. a JKM 2. a J 47. The triangles are not congruent. 4. a KML 5. a LKM 48. Perpendicular lines intersect to form four right angles, 6. Yes; T ABC T DEC by the SAS Congruence Postulate and all right angles are congruent. 49. By the Base Angles Theorem, the base angles of an isosceles triangle are congruent, so a JGF a JHF. 50. Yes; by the Reflexive Property of Congruence, FJ **** FJ **** so all three pairs of corresponding sides are congruent. By Exercises 48 and 49, all three pairs of corresponding angles are congruent. 51. a BAC and a DAE are congruent because of the Reflexive Property of Congruence. 52. No; the corresponding angles are congruent, but the corresponding sides are not congruent. That is, ED AB . **** ç CB **** , AE **** ç AC **** , and AD **** ç **** 5.1 Standardized Test Practice (p. 239) 53. B 5.1 Mixed Review (p. 239) 54. Reflexive Property of Equality 55. Reflexive Property of Congruence 56. Symmetric Property of Congruence 57. Transitive Property of Equality 70 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 3. a L because **** AC DC **** , a ACB a DCE, and BC **** EC **** . 7. No; there is nothing in the diagram indicating that a FHG a JHK or that FG **** JK **** . 8. Yes; T PQR T SRQ by the SSS Congruence Postulate. By the Reflexive Property of Congruence, RQ **** RQ **** , so all three pairs of corresponding sides are congruent. 5.2 Practice and Applications (pp. 245–249) 9. a ABD 12. a A 10. a C 13. a BDC 11. a C 14. a DBC 15. Yes; T ABC T DBE because all three pairs of corre- sponding sides of the triangles are congruent. 16. Yes; by the Reflexive Property of Congruence, **** JL **** JL so T JKL T JML because all three pairs of corresponding sides of the triangles are congruent. 17. No; there is no information about PS **** and RS ****. 18. Yes; by the Reflexive Property of Congruence, SQ **** SQ **** so T PQS T RQS because two sides and the included angle of T PQS are congruent to two sides and the included angle of T RQS. 19. No; the congruent angles are not included by the congruent sides. Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 20. Yes; by the Vertical Angles Theorem, a JNK a MNL so T JNK T MNL because two sides and the included angle of T JNK are congruent to two sides and the included angle of T MNL. 21. No; the congruent angles are not included by the congruent sides. 22. Yes; T GHK T JKH by the SAS Congruence Postulate because GH **** JK **** (Given), a GHK a JKH (Alternate Interior Angles Theorem), and KH **** KH **** (Reflexive Property of Congruence). 23. Yes; T MNP T PQM by the SSS Congruence Postulate because MN **& PQ **** (Given), **& MQ PN **** (Given), and **& (Reflexive Property of Congruence). MP **& MP 24. Yes; T WXZ T YXZ by the SAS Congruence Postulate because WX XZ **** XZ (Reflexive **** YX **** (Given), and **** Property of Congruence). 36. Statements 1. **** AC BC **** AB and CB 26. No; there is no information about **** ****. AB CD 27. **** **** 28. a ACB a CED 29. To use the SAS Congruence Postulate, you need to know that BC **** EF ****. 30. To use the SSS Congruence Postulate, you need to know that JK **** ML **** . To use the SAS Congruence Postulate, you need to know that a JLK a MKL. 31. To use the SSS Congruence Postulate, you need to know that PQ **** RQ **** . To use the SAS Congruence Postulate, you need to know that a PSQ a RSQ. 32. Sample answer: Since WXYZ is a square, all four sides of WXYZ are congruent. **** XZ **** XZ by the Reflexive Property of Congruence, so TWXZ TYXZ by the SSS Congruence Postulate. AB CD 33. Sample answer: Since **** **** , a ABC a DCB AB DC (Alternate Interior Angles Theorem). **** **** (Given) and CB **** CB **** (Reflexive Property of Congruence), so T ABC T DCB by the SAS Congruence Postulate. 1. Given 2. M is the midpoint of **** AB. 2. Given 3. AM **& BM **** 3. Definition of midpoint 4. CM **& CM **& 4. Reflexive Prop. of Cong. 5. T ACM T BCM 5. SSS Cong. Postulate 37. Maria should not have used the SAS Congruence Postulate because the congruent angles are not included by the congruent sides. 38. Statements Reasons 1. T ABD and T CBD are equilateral. 1. Given 2. BD **** BD **** 2. Reflexive Prop. of Cong. 3. BC **** BD **** , 3. Definition of equilateral 25. Yes; T RSU T TSU by the SSS Congruence Postulate because RS **** TS **** (Given), RU **** TU **** (Given), and SU **** SU **** (Reflexive Property of Congruence). Also, T RSU T TSU by the SAS Congruence Postulate because RS **** TS **** (Given), a SRU a STU (Base Angles Theorem), and RU **** TU **** (Given). Reasons BA BD **** **** 4. BC BA **** **** 4. Transitive Prop. of Cong. 5. DC **** BD **** , 5. Definition of equilateral DA **** BD **** 6. DC **** DA **** 7. T ABD T CBD 39. Statements 6. Transitive Prop. of Cong. 7. SSS Cong. Postulate Reasons 1. XZ **** YZ **** 1. Given 2. **& ZM bisects a YZX. 2. Given 3. a YZM a XZM 3. Def. of angle bisector 4. **& ZM **& ZM 4. Reflexive Prop. of Cong. 5. T ZMY T ZMX 5. SAS Cong. Postulate 5.2 Standardized Test Practice (p. 248) 40. C 41. G 5.2 Mixed Review (p. 248) 42. alternate exterior angles 43. same-side interior angles 44. corresponding angles 45. alternate interior angles 46. corresponding angles 47. same-side interior angles Reasons 48. no; 14 8 25 1. EF **** GH **** 1. Given 49. yes; 20 10 28, 20 28 10, and 10 28 20 2. FG **** HE **** 2. Given 50. no; 16 14 30 3. GE **** GE **** 3. Reflexive Prop. of Cong. 5.2 Algebra Skills (p. 249) 4. T EFG T GHE 4. SSS Congruence Postulate 34. Statements 35. Statements Reasons 1. SP **** TP **** 1. Given 2. PQ **** bisects a SPT. 2. Given 3. a SPQ a TPQ 3. Def. of angle bisector 4. PQ **** PQ **** 4. Reflexive Prop. of Cong. 5. T SPQ T TPQ 5. SAS Cong. Postulate Copyright © McDougal Littell Inc. All rights reserved. 1.7 51. 3 52. 1 2 3.5 53. 4 0 6.3 54. 1 5 9 12.6 55. 1 4 .7 6 3.8 56. 0 .8 7 0.9 57. 1 .1 2 1.1 58. 4 0 .8 5 6.4 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 71 Chapter 5 continued 11. **** AC is included between a BAC and a BCA. Quiz 1 (p. 249) 1. a R a C 2. **** AB QP **** 12. **** AC is not included between a DAC and a D. 3. T BAC T PQR 4. T RPQ T CBA 13. BC **** is not included between a CAB and a B. 5. T FEG T XYZ, T EGF T YZX, or T GFE T ZXY 6. No; the congruent angles are not included by the congruent sides. 7. Yes; by the Reflexive Property of Congruence, FG **** FG **** . The three sides of T DGF are congruent to the three sides of T EGF, so T DGF T EGF by the SSS Congruence Postulate. 8. Yes, two sides and the included angles of T JKF are congruent to two sides and the included angle of T NMF, so the triangles are congruent by the SAS Congruence Postulate. 14. ASA Congruence Postulate; two angles and the included side of T ABC are congruent to two angles and the included side of T DEF. 15. AAS Congruence Theorem; two angles and a non- included side of T JLM are congruent to two angles and the corresponding non-included side of T PNM. 16. ASA Congruence Postulate; two angles and the included side of T URS are congruent to two angles and the included side of T UTS. 17. Yes; SAS Congruence Postulate; two sides and the included angle of T PQR are congruent to two sides and the included angle of T TSM. 18. Yes; AAS Congruence Theorem; a A a D (Given), Lesson 5.3 5.3 Geo-Activity (p. 250) Step 5. The triangles are congruent. a BCA a BCD (Perpendicular lines form right angles, and all right angles are congruent.), and BC **** BC **** (Reflexive Property of Congruence). So, T BCA T BCD. 19. Yes; ASA Congruence Postulate; a S a T (Given), 5.3 Checkpoint (p. 253) 1. You can use the ASA Congruence Postulate when the corresponding sides are included between the corresponding angles. 2. No; based on the diagram, you know only that a BAC a DCA (Alternate Interior Angles Theorem) and **** AC **** AC . 3. No; there is no information about any of the sides. 4. Yes; T KLP T MLN by the AAS Congruence Theorem because a P a N (Given), a KLP a MLN (Vertical Angles Theorem), and KL **** ML **** (Given). 5.3 Guided Practice (p. 253) 1. FG **** is included between a F and a G. 2. GH **** is not included between a F and a G. SV **** TV **** (Given), and a SVR a TVU (Vertical Angles Theorem). So, T SVR T TVU. 20. No; from the diagram, you know only that EF JH , **** **** FG **** HG **** , and a EGF a JGH (Vertical Angles Theorem). The congruent angles are not included by the congruent sides. 21. Yes; SSS Congruence Postulate; WX ZX (Given), **& **** WY XY **** XY (Reflexive Property of **& ZY **** (Given), and **** Congruence). So, T XWY T XZY. 22. No; from the diagram, you know only that NL **** NL **** (Reflexive Property of Congruence) and a MNL a KLN (Alternate Interior Angles Theorem). 23. AAS Congruence Theorem; two angles and a non- included side of T JKL are congruent to two angles and the corresponding non-included side of T PQR. 3. FH **** is not included between a H and a G. 24. AC **** FD **** 25. a K a Q 4. HG **** is included between a H and a G. 27. a C a D 28. a CAB a DAB 5. **** AB DE **** 30. BC **** BD **** 6. a A a D 7. Yes; T RST T TQR by the ASA Congruence Postulate 26. a WZX a YXZ 29. AC **** AD **** 31. Sample answer: because a STR a QRT (Given), RT **** RT **** (Reflexive Property of Congruence), and a SRT a QTR (Given). T 8. Yes; T JKL T NML by the AAS Congruence Theorem because a KLJ a MLN (Vertical Angles Theorem), a JKL a NML (Given), and **** JK NM **& (Given). 9. No; you would either need to know that a A a D or P R S U 32. Sample answer: a C a F. 5.3 Practice and Applications (pp. 254–256) T P R S U AB is not included between a B and a BCA. 10. **** 72 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 7 9 P 2 7 7 9 7 7 9 2 7 2 49 18 13 18 50. 2 33. Sample answer: S 12 25 3 10 3 10 25 12 1 3 25 5 2 10 12 4 5 8 51. R T U 34. Statements 3 4 4 3 8 4 1 3 1. GF **** GL **** 1. Given 2. FH **** LK **** 2. Given 3. a F a L 3. Alt. Int. Angles Theorem 4. a HGF a KGL 4. Vertical Angles Theorem Lesson 5.4 5. T FGH T LGK 5. ASA Cong. Postulate 5.4 Checkpoint (p. 259) 35. Statements Reasons 1. BC **** EC **** 32 3 2 3 52. 8 8 10 Reasons 4 11 4 11 1 12 1 4 1 11 12 3 1 33 53. 12 1. Yes; HL Congruence Theorem; the hypotenuse and a leg 2. **** AB ∏ AD **** 2. Given of right T BAC are congruent to the corresponding hypotenuse and a leg of right T DEC. 3. DE ∏ AD 3. Given 2. No; from the diagram, you know only that SU **** SU **** 4. a A and a D are right angles. 4. Perpendicular lines form right angles. 5. a A a D 5. Right angles are congruent. 6. a ACB a DCE 6. Vertical Angles Theorem 7. T ABC T DEC 7. AAS Congruence Theorem 1. Given 5.3 Standardized Test Practice (p. 256) (Reference Property of Congruence) and a RSU a TUS (Alternate Interior Angles Theorem). 3. Yes; SSS Congruence Postulate, SAS Congruence Postulate, or HL Congruence Theorem 5.4 Guided Practice (p. 260) 1. hypotenuse 2. leg 3. leg 4. leg 36. D 5. hypotenuse 6. leg 5.3 Mixed Review (p. 256) 7. Yes; all three sides of T EDG are congruent to the three 37. 102 b2 182 38. c2 112 62 100 b2 324 c2 121 36 b2 224 c2 157 b 224 c 157 b 15.0 c 12.5 39. 15 a 26 2 2 2 a2 451 gruent to the hypotenuse and a leg of right T FED. 40. a C a F 41. BA **** ED **** 42. a D a A 43. EF **** BC **** 44. a B a E 45. CB **** FE **** 5.3 Algebra Skills (p. 256) 5 16 1 3 26 2 3 2 1 1 4 48. 5 8 16 5 11. Yes; **** JL **** JL (Reflexive Property of Congruence), so the hypotenuse and a leg of right T JKL are congruent to the hypotenuse and a leg of right T JML. 12. No; there is no information given about the lengths of the 1 1 1 1 1 1 46. 6 12 2 2 6 2 6 4 4 1 4 1 1 47. 4 5 5 4 5 4 5 5 8 9. No; both triangles are equilateral and equiangular, but 10. Yes; the hypotenuse and a leg of right T ABC are con- a 21.2 1 6 congruent to the hypotenuse and a leg of right T QPM, so the triangles are congruent by the HL Congruence Theorem. 5.4 Practice and Applications (pp. 260–263) a 451 2 3 8. Yes; the hypotenuse and a leg of right T MNQ are there is no information about the lengths of the sides of the triangles. 225 a2 676 1 6 sides of T GFE so the triangles are congruent by the SSS Congruence Postulate. hypotenuses of the right triangles. 13. **** JL **& ML 14. SAS Congruence Postulate; since K is the midpoint of JM **** , JK **** **&. MK It is given that a JKL and a MKL are right angles, and all right angles are congruent. By the Reflexive Property of Congruence LK **** LK ****. 1 5 16 2 1 8 51 49. 2 Copyright © McDougal Littell Inc. All rights reserved. Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 73 Chapter 5 continued 15. Yes; SAS Congruence Postulate; two sides and the 27. Sample answer: included angle of T ABC are congruent to two sides and the included angle of T DEF. N U 16. Yes; ASA Congruence Postulate; two angles and the included side of T JKL are congruent to two angles and the included side of T ZYX. 17. No; the congruent angles are not included between the congruent sides, so there is not enough information to prove that the triangles are congruent. 18. Yes; SSS Congruence Postulate; FH **** FH **** (Reflexive Property of Congruence), so all three sides of T FGH are congruent to the three sides of T FJH. 19. Yes; AAS Congruence Theorem; perpendicular lines form right angles, and all right angles are congruent, so a UTS a UTV. Also, by the Reflexive Property of Congruence, UT **** UT **** . Two angles and a non-included side of T STU are congruent to two angles and the corresponding non-included side of T VTU. 20. Yes; AAS Congruence Theorem; perpendicular lines form right angles, and all right angles are congruent, so a A a D. Also, vertical angles are congruent, so a ACB a DCE. Two angles and a non-included side of T ABC are congruent to two angles and the corresponding non-included side of T DEC. 21. Yes; HL Congruence Theorem; by the Reflexive Property of Congruence, DB **** DB **** , so the hypotenuse and a leg of right T ADB are congruent to the hypotenuse and corresponding leg of right T CDB. 22. Yes; ASA Congruence Postulate; a JKL a MLK and a JLK a MKL by the Alternate Interior Angles Theorem. LK **** LK **** by the Reflexive Property of Congruence. So, two angles and the included side of T JKL are congruent to two angles and the included side of T MLK. 23. Yes; SAS Congruence Postulate; DF **** DF **** (Reflexive Property of Congruence), so two sides and the included angle of T CDF are congruent to two sides and the included angle of T EDF. 24. Meghan is correct because it is given that AB **** CD **** and BC **** DA **** , and by the Reflexive Property of Congruence, AC **** AC . Keith is correct because two sides and the **** included angle of T ABC are congruent to two sides and the included angle of T CDA. Angle is correct because the hypotenuse (AC **** ) and a leg of right T ABC are congruent to the hypotenuse and a leg of right T CDA. 25. Sample answer: L T N U M S 26. Sample answer: L 74 L M T S 28. Sample answer: M T L N S U 29. a F a J, ASA Congruence Postulate; or a G a L, AAS Congruence Theorem; or GH **& LK **** , SAS Congruence Postulate. 30. PQ **& RS **** or PS **** RQ **** , HL Congruence Theorem; or a PQS a RSQ or a PSQ a RQS, AAS Congruence Theorem 31. WV **& ZX **** , ASA Congruence Postulate; or VW **& YZ **** , or VX **** YX **** , AAS Congruence Theorem 32. Statements Reasons 1. BD **** FD **** 1. Given 2. D is the midpoint of CE **** . 2. Given 3. CD **** ED **** 3. Definition of midpoint 4. a BCD and a FED are right angles. 4. Given 5. T BCD and T FED are right triangles. 5. Definition of right triangle 6. T BCD T FED 6. HL Congruence Theorem 5.4 Standardized Test Practice (p. 262) 33. a. Sample answer: BD **** BD **** by the Reflexive Property of Congruence; the hypotenuse and a leg of right T ABD are congruent to the hypotenuse and a leg of right T CBD, so T ABD T CBD by the HL Congruence Theorem. b. Yes; Sample answer: It is given that AB **** BC **** . By the Base Angles Theorem, a BAD a BCD. Perpendicular lines form right angles, and all right angles are congruent, so a BDA a BDC. Therefore, T ABD T CBD by the AAS Congruence Theorem. c. Yes; Sample answer: You know that T CEG T FEG by the HL Congruence Theorem. It is given that the hypotenuse and a leg of right T ABD are to the hypotenuse and a leg of right T CEG. So, TABD TCEG by the HL Congruence Theorem. Since TABD TCBD, all four right triangles are congruent. 5.4 Mixed Review (p. 263) N U M T 34. ma 1 110 by the Alternate Interior Angles Theorem; S Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key ma 2 180 110 70 by the Same-Side Interior Angles Theorem (or by the Linear Pair Postulate). Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 35. ma 1 82 by the Corresponding Angles Postulate; 10. Yes; SSS Congruence Postulate; T JML and T JKL are ma 2 82 by the Alternate Exterior Angles Theorem (or by the Vertical Angles Theorem). equiangular and therefore equilateral. Because they share a common side (JL **** ) you know that all the sides of T JML are congruent to all the sides of T JKL; or AAS Congruence Theorem because **** JL **** JL (Reflexive Property of Congruence), two angles and a non-included side of T JML are congruent to two angles and the corresponding non-included side of T JKL; or ASA Congruence Postulate because **** JL **** JL , two angles and the included side of T JML are congruent to two angles and the included side of T JKL. 36. ma 1 180 57 123 by the Linear Pair Postulate; ma 2 57 by the Alternate Interior Angles Theorem. 37. Yes; AAS Congruence Theorem; two angles and a non- included side of T ABC are congruent to two angles and a non-included side of T DEF. 38. No; a PQS a RSQ by the Alternate Interior Angles Theorem and SQ **** SQ **** by the Reflexive Property of Congruence, but the congruent angles are not included between the congruent sides. 39. Yes; AAS Congruence Theorem; a JLK a NLM by the Vertical Angles Theorem, so two angles and a nonincluded side of T JKL are congruent to two angles and the corresponding non-included side of T NML. 5.4 Algebra Skills (p. 263) 40. 2 4 5 8 5 13 5.4 Technology Activity (p. 264) 1. Answers will vary, but BG BH. 2. **** AB **** AB and BG **** BH **** 3. a BAG a BAH 4. No; Sample answer: T ABH and T ABG have different shapes. 5. 41. 10 5 2 10 10 0 42. 3 42 11 3 16 11 19 11 8 43. 7 2 6 3 14 18 32 A NM **& PM **& (Given), and a LNM a QPM (All right angles are congruent.). 2. Yes; a NML a PMQ (Vertical Angles Theorem), H Lesson 5.5 5.5 Checkpoint (p. 267) 1. SAS Congruence Postulate; AD **** CB **** (Given), a DAC a BCA (Alternate Interior Angles Theorem), and **** AC **** AC (Reflexive Property of Congruence). So, T ADC T CBA and the corresponding parts **** AB and CD **** are congruent. a LNM a QPM (All right angles are congruent.) and LN **** QP **** (Given). 3. No; no information is given about the lengths of the hypotenuses. A sible for two sides and a non-included angle of one triangle to be congruent to two sides and the corresponding non-included angle of another triangle without the two triangles being congruent. 45. 52 10 2 25 20 5 1. Yes; a NML a PMQ (Vertical Angles Theorem), G 6. There is no SSA Congruence Postulate because it is pos- 44. 3 5 2 7 15 14 1 Quiz 2 (p. 263) B B 2. K K 4. No; no information is given about the third sides. N 5. No; the congruent sides are not corresponding sides. 6. Yes; AAS Congruence Theorem; two angles and a non- J included side of T JKL are congruent to two angles and the corresponding non-included side of T QRP. 8. Yes; HL Congruence Theorem; the hypotenuse and a leg of right T HEF are congruent to the hypotenuse and a right leg of T FGH. 9. Yes; AAS Congruence Theorem; a RTS a UTV L T JKN T LKM by the ASA Congruence Postulate, so the corresponding parts NJ ML are congruent. **** and **& 7. Yes; ASA Congruence Postulate; TU **** TU **** by the Reflexive Property of Congruence, so two angles and the included side of T STU are congruent to two angles and the included side of T VUT. M 3. S P R P R In the original diagram, PR **** and PR **** are the same side, so PR **** PR **** . T PQR T RSP by the AAS Congruence Theorem. (Vertical Angles Theorem), a RST a VUT (Alternate Interior Angles Theorem), and RS **** UV **& (Given). So, two angles and a non-included side of T RST are congruent to two angles and the corresponding non-included side of T UVT. Copyright © McDougal Littell Inc. All rights reserved. Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 75 Chapter 5 continued 5.5 Guided Practice (p. 268) 15. Statements Reasons 1. Corresponding parts of congruent triangles are congruent. 1. AD **** CD **** 1. Given 2. B 2. a ABD and a CBD are right angles. 2. Given 3. T ABD T CBD are right triangles. 3. Def. of right triangle 4. BD **** BD **** 4. Reflexive Property of Cong. 5. T ABD T CBD 5. HL Congruence Theorem 6. a A a C 6. Corresponding parts of triangles are . 3. When two parallel lines are cut by a transversal, alternate interior angles are congruent, so a VSU a TUS and a TSU a VUS. Also, SU **** SU **** by the Reflexive Property of Congruence. T STU T UVS by the ASA Congruence Postulate and a STU a UVS since corresponding parts of congruent triangles are congruent. 4. a A a D, a DBC a ACB, and BC **** BC **** . So, T ABC T DCB by the AAS Congruence Theorem and **** AB DC **** since corresponding parts of congruent triangles are congruent. 5. LN **** LK **** , a LNM a LKJ, and a L a L. So, T JKL T MNL by the ASA Congruence Postulate and a J a M since corresponding parts of congruent triangles are congruent. 5.5 Practice and Applications (pp. 268–271) 6. Corresponding parts of congruent triangles are congruent. 7. T ABC T DBC 8. T JKL T NML 9. T EDF T ABC 10. B D A C In the original diagram, DB **** and DB **** are the same side, so DB **** DB **** . T ABD T CDB by the SAS Congruence Postulate. 11. E J G G 12. AAS Congruence Theorem; a C a F, a CBA a FED, and CA **** FD **** , so T ABC T DEF. 13. ASA Congruence Postulate; a GJH a KMH and HJ **** HM **&. By the Reflexive Property of Congruence, a H a H. So, T GHJ T KHM. 14. Statements 1. Given 2. a JKL a NML 2. Alternate Interior Angles Theorem 3. KL **** ML **** 3. Given 4. a JLK a NLM 4. Vertical Angles Theorem 5. T JKL T NML 5. ASA Congruence Postulate 6. **** JK NM **& 6. Corresponding parts of triangles are . Reasons Reasons 1. a TRS a RTQ 1. Given 2. a RST and a TQR are right angles. 2. Given 3. a RST a TQR 3. All right angles are . 4. RQ **** TS **** 4. Given 5. T RQT T TSR 5. AAS Congruence Theorem 6. QT **** SR **** 6. Corresponding parts of triangles are . H In the original diagram, a G and a G are the same angle, so a G a G. T EFG T HJG by the AAS Congruence Theorem. Reasons 1. JK NM **** **& 17. Statements D B F 16. Statements 18. Statements Reasons 1. UV XY **** **** 1. Given 2. a UVW a XYZ 2. Given 3. VW **& YZ **** 3. Given 4. T UVW T XYZ 4. SAS Congruence Postulate 5. WU **& ZX **** 5. Corresponding parts of triangles are . 19. Statements Reasons 1. **** AB AE **** 1. Given 2. a ACB a ADE 1. BD **** and AE **** bisect each other at C. 1. Given 2. Given 3. a A a A 3. Reflexive Prop. of Cong. 2. BC **** DC **** 2. Def. of segment bisector 4. T ABC T AED 4. AAS Congruence Theorem 3. **** AC EC **** 3. Def. of segment bisector 5. a B a E 5. Corresponding parts of triangles are . 4. a BCA a DCE 4. Vertical Angles Theorem 5. T ABC T EDC 5. SAS Congruence Postulate 6. a A a E 6. Corresponding parts of triangles are . 76 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 20. Statements Reasons 1. **** JL **& MK 1. Given 2. JK **** ∏ LK **** , ML **** ∏ KL **** 2. Given 3. a JKL and a MLK are right angles. 3. Perpendicular lines form right angles. 4. T JKL and T MLK are right triangles. 4. Definition of right triangle 5. KL **** KL **** 5. Reflexive Prop. of Cong. 6. T JKL T MLK 6. HL Congruence Theorem 7. JK **** ML **** 7. Corresponding parts of triangles are . 21. ASA Congruence Postulate; a LJK a PMN (Given), JL **& MP (Segment Addition Postulate), and a JLK **** a MPN (Corresponding Angles Theorem). Two angles and the included side of T JKL are congruent to two angles and the included side of T MNP. 29. 8x 2 2 90 2 8x 88 8x 88 8 8 x 11 5.5 Algebra Skills (p. 271) 30. x3 31. 7x 63 7x 63 7 7 x 9 32. 4x 32 4x 32 4 4 x8 23. G (x 3) 25 x 3 3 25 3 4x 9 23 4x 9 9 23 9 5.5 Mixed Review (p. 271) 24. x58 x5585 5.5 Standardized Test Practice (p. 271) 22. A (8x 2) 90 3x 8 19 2x 10 10 24 10 3x 8 8 19 8 2x 14 2x 14 2 2 x7 (6x 1) (5x 5) x1151 x6 27. x 48 90 28. (4x 1) 37 90 5.6 Activity (p. 272) 1. CM ^&( is perpendicular to **** AB and CM ^&( bisects **** AB . 2. Answers will vary, but CA CB, DA DB, EA EB, and FA FB. 3. Any point on the perpendicular bisector of a segment is the same distance from the endpoints of the segment. 5.6 Checkpoint (p. 274) 1. FH HG 4x 38 90 x 3 2x 1 4x 38 38 90 38 x 3 x 2x 1 x 4x 52 4x 52 4 4 x 13 Copyright © McDougal Littell Inc. All rights reserved. 3x 27 3x 27 3 3 x9 Lesson 5.6 x 48 48 90 48 x 42 3x 21 3x 21 3 3 x7 2x 10 24 3x 54 3 3 x 18 x15 11 3x 11 32 11 35. x 2x 8 19 25. 3x 54 6x 1 5x 5x 5 5x 11 3x 32 34. 5x 3x 10 24 x 28 26. 33. 3x1 31x11 2x FH x 3 2 3 5 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 77 Chapter 5 continued 9. Paige cannot assume that PC & *(. **** ∏ AC LM MK 2. 4x x 15 10. x 10 12. HE HF 3x x 4 MK x 15 5 15 20 3x x x 4 x EF GF 2x 4 2x 4 2 x2 2x 5 x 10 2x 5 x x 10 x x 5 10 13. BC **** BD **** by the Angle Bisector Theorem. x 5 5 10 5 14. ma JKM ma LKM 38 x5 15. SV UV 18 EF 2x 5 2(5) 5 10 5 15 17. 5.6 Guided Practice (p. 276) LK NK x x 2x 3 x equidistant from the two sides of the angle. 0x3 2. If D is on the perpendicular bisector of **** AB , then D is 03x33 equidistant from A and B. 3x 3. AD DC 16 LK x 3 EF FG 18. x 1 2x 1 PQ RQ x 2 2x 1 x 1 x 2x 1 x x 2 x 2x 1 x 1x1 2x1 11x11 21x11 2x 3x EF x 1 2 1 3 PQ x 2 3 2 5 5. JM ML 12 19. AD CD 12; QR QP 3x 8 5x BC BA 3x 8 3x 5x 3x 2x 6 3x 1 2x 6 2x 3x 1 2x 8 2x 8 2x 2 2 4x 6x1 61x11 5x QR 3x 8 3(4) 8 12 8 20 BC 2x 6 2(5) 6 10 6 16 20. Perpendicular Bisector Theorem; Because DB **** is perpen- 5.6 Practice and Applications (pp. 276–280) 7. dicular to and bisects **** AC , point D is on the perpendicular bisector of **** AC . So, D is equidistant from points A and C. 8. A C A B D C B D 78 16. HG FG 14 x 2x 3 1. If a point is on the bisector of an angle, then it is 6. x6 x 5 5 15 5 3x 15 3x 15 3 3 x5 4. 11. MJ ML x 5 15 4x x x 15 x 3. CB CA Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 21. Line l is the perpendicular bisector of the goal line (AC **** ). 22. BG &*( should bisect a ABC so that the goalkeeper is equidistant from the edges of the region that the goalkeeper is defending. By doing this, the goalkeeper minimizes the distance needed to reach the ball. Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 23. CG AG BG 2 24. VR VQ VP 16 25. GL **** LH **** ; 5.6 Standardized Test Practice (p. 280) 34. B; SR SP FJ JG ; FK **** **** **** KH **** ; x 2x 3 FM GM HM **** **& *& *** 26 –28. x x 2x 3 x 0x3 A 03x33 (red) 3x X (blue) (gray) SR x 3 B 35. G; C 29. The fire station at point A; X is in the red region, which 2x 3 x x x x30 is the region closest to station A. 30. DA **** DB **** ; the measures of DA **** and DB **** are equal. x3303 x3 31. Yes; by the Perpendicular Bisector Theorem, if D lies on the perpendicular bisector of **** AB , then DA **** and DB **** will always be congruent segments. 32. Statements SP RS 2x 3 x Reasons SP 2x 3 2(3) 3 6 3 3 36. D 1. ^&( AD is the perpendicular bisector of BC **** . 1. Given 2. DB **** DC **** 2. Definition of perpendicular bisector 3. a ADB and a ADC are right angles. 3. Perpendicular lines form right angles. 4. a ADB a ADC 4. Right angles are . 41. (6, 2) → (6 3, 2 6) (9, 4) 5. AD **** AD **** 5. Reflexive Prop. of Cong. 42. (2, 5) → (2 3, 5 6) (5, 11) 6. T ADB T ADC 6. SAS Cong. Postulate 43. (10, 12) → (10 3, 12 6) (13, 6) 7. **** AB **** AC 7. Corresponding parts of triangles are . 44. (1, 1) → (1 3, 1 6) (2, 7) 8. AB AC 8. Def. of segments 33. Statements Reasons 5.6 Mixed Review (p. 280) 37. (5, 1) → (5 3, 1 6) (8, 5) 38. (2, 3) → (2 3, 3 6) (1, 3) 39. (4, 4) → (4 3, 4 6) (1, 10) 40. (0, 6) → (0 3, 6 6) (3, 12) 45. SAS Congruence Postulate; DB **** DB **** (Reflexive Property of Congruence) and all right angles are congruent, so two sides and the included angle of T ADB are congruent to two sides and the included angle of T CDB. 1. D is on the bisector of a BAC. 1. Given 2. a BAD a CAD 2. Definition of angle bisector 3. DB &*( and **** ∏ AB DC AC *( **** ∏ & 3. Given 4. a DBA and a DCA are right angles. 4. Perpendicular lines form right angles. 5. a DBA a DCA 5. Right angles are . 5.6 Algebra Skills (p. 280) 6. DA **** DA **** 6. Reflexive Prop. of Cong. 7. T ADB T ADC 48. 3, 0.3, 0, 0.3, 0.6, 3 7. AAS Cong. Theorem 8. DB **** DC **** 49. 1.25, 0.75, 0.25, 0.25, 1, 4 8. Corresponding parts of triangles are . 50. 4, 0.4, 0.1, 0, 0.1, 4.0 Copyright © McDougal Littell Inc. All rights reserved. 46. AAS Congruence Theorem; EF **** EF **** (Reflexive Property of Congruence), so two angles and a nonincluded side of T HEF are congruent to two angles and the corresponding non-included side of T GFE. 47. ASA Congruence Postulate; vertical angles are congruent so two angles and the included side of T JNK are congruent to two angles and the included side of T LNM. 51. 3.9, 3.3, 3, 3.1, 3.5, 3.8 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 79 Chapter 5 continued 52. 1, 0.1, 0, 0.5, 0.55, 1, 1.1 62. 53. 2.5, 1, 2.1, 3.2, 3.25, 5 4x 3 11 54. 10t 10 10t 12t 10t 55. 4x 3 3 11 3 4x 8 4x 8 4 4 x2 2y 9 11 2y 2 2y 2 2 2 y 1 5d 35 35 90 35 5d 125 5d 125 5 5 d 25 57. 4a 9a 39 13a 39 13a 39 13 13 a3 x 2 3x 4 58. 10 2t 10 2t 2 2 5t 2y 9 9 11 9 5d 35 90 56. 10t 10 12t x 2 x 3x 4 x Lesson 5.7 5.7 Activity (p. 281) XA **& AX, ZB 1. Answers will vary, but **** **** BZ **&, and YC *& CY **&. XA **& AX, ZB 2. **** **** BZ **&, and YC *& CY **&; each pair of segments is congruent. 3. 90; 90; 90; the measure of each angle is 90. 4. The fold line is the perpendicular bisector of each segment. 5.7 Checkpoint (pp. 283–285) 1. yes; the x-axis 2. no 4. The figure has one line of symmetry. 3. yes; the y-axis 5. The figure has two lines of symmetry. 2 2x 4 2 4 2x 4 4 6 2x 6 2x 2 2 3x 6. The figure has four lines of symmetry. 4r 2 5r 6 59. 4r 2 4r 5r 6 4r 2 r 6 2 6 r 6 6 8 r q 2q 9 60. q 2q 2q 9 2q q 9 q 9 1 1 q9 2z 5 4z 1 61. 2z 5 2z 4z 1 2z 5.7 Guided Practice (p. 286) 1. A figure in the plane has a line of symmetry if the figure can be reflected onto itself by a reflection in the line. 2. No; the orientation of the image is not reversed. 3. Yes; all three properties of a reflection are met. 4. Yes; all three properties of a reflection are met. 5. The flower has three lines of symmetry. 6. The flower has two lines of symmetry. 5 2z 1 5 1 2z 1 1 6 2z 6 2z 2 2 3z 80 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 7. The flower has five lines of symmetry. 22. yes 23. yes 24. The figure has five 25. The figure has four lines of symmetry. lines of symmetry. 26. The figure has two lines of symmetry. 5.7 Practice and Applications (pp. 286–290) 8. Yes; all three properties of a reflection are met. 9. Yes; all three properties of a reflection are met. 10. No; the orientation is not reversed. 27. All the lines of symmetry are shown. 11. The grid shows a reflection in the y-axis. 28. no; 29. no; 12. The grid shows neither a reflection in the x-axis, nor a reflection in the y-axis. 13. The grid shows a reflection in the x-axis. 14. GH AB in the x-axis; point G **** is a reflection of **** corresponds to A; point H corresponds to B. AB in the y-axis; point C 15. CD **** is a reflection of **** corresponds to A; point D corresponds to B. 16. Coordinates of **** AB : A(3, 1), B(1, 3). Coordinates of GH **** : G(3, 1), H(1, 3). The coordinates are alike in that the x-coordinate of each original point is the same as the x-coordinate of the image point. The coordinates are different in that the y-coordinate of each original point is the opposite of the y-coordinate of the image point. 17. 18. 30. 31. The letters b and d are reflections of each other, and p and q are reflections of each other. k 32. One line of symmetry: , , , and ; k Two lines of symmetry: 19. , , 33. yes; k 34. no 35. yes; 36. yes; 20. All three triangles in sections B, and D are reflections of the triangle in section A. Each fold in the paper is a line of symmetry. 21. No; the right side of the guitar is not the same as the left side. Copyright © McDougal Littell Inc. All rights reserved. Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 81 Chapter 5 continued 180 n 180 4 45 180 n 180 2 90 37. ma A 38. ma A 180 n 180 3 60 48. ma 1 51 30 180 ma 1 81 180 ma 1 81 81 180 81 ma 1 99 ma 1 44 90 180 49. 39. ma A ma 1 134 180 ma 1 134 134 180 134 ma 1 46 40. AB ⏐3 1⏐ 2, DE ⏐3 (1)⏐ ⏐2⏐ 2; AC ⏐6 2⏐ 4, DF ⏐6 2⏐ 4; 2 )2 (1 3 )2 4 2( 2 )2 2 0 , BC (6 EF (6 2 )2 ( 1 ( 3)) 2 4222 20; By the SSS Congruence Postulate, T ABC T DEF. 41. AB ⏐1 3⏐ ⏐2⏐ 2, DE ⏐1 (3)⏐ ⏐2⏐ 2; AC ⏐4 1⏐ 3, DF ⏐4 (1)⏐ ⏐3⏐ 3; 5.7 Algebra Skills 50. 2348 2384 51. 5 7 52. 19.1 19.01 53. 11.2 11.238 54. 0.065 0.056 55. 1.011 1.11 Quiz 3 (p. 290) 1. A 1 )2 (4 1 )2 2 232 13, BC (3 B EF ( 4 ( 1)) 2( 3 ( 1)) 2 ( 3 )2 ( 2 )2 1 3 ; By the SSS Congruence Postulate, T ABC T DEF. C B C AAS Congruence Postulate 2. DC AD 2 3. ML MJ 9; JK LK 25 5.7 Standardized Test Practice (p. 289) 4. 42. C D AB CD 3x 4 2x 7 43. G; 3x 4 2x 2x 7 2x x47 x4474 5.7 Mixed Review (p. 290) 44. x 105 x3 AB 3x 4 3(3) 4 13 5. The figure has one line of symmetry. x 105 (x 10) 82 45. x 10 10 82 10 x 72 (3x 1) 92 46. 3x 1 1 92 1 6. The figure has two lines of symmetry. 7. The figure has three lines of symmetry. 3x 93 3x 93 3 3 x 31 ma 1 38 75 180 47. ma 1 113 180 Chapter 5 Summary and Review (pp. 291–295) ma 1 113 113 180 113 1. When two figures are congruent, their corresponding ma 1 67 82 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key sides and their corresponding angles are congruent. Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 2. A proof is a convincing argument that shows why a 23. Statements statement is true. 1. a JMN and a JKL are right angles. 3. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. 4. If a point is on the angle bisector of an angle, then it is equidistant from the two sides of the angle. 5. A reflection is a transformation that creates a mirror image. 6. neither 7. corresponding sides 8. corresponding angles 9. neither 10. neither 11. corresponding angles because it is given that DE AC , a E a C, and **** **** EF **** CB **** . 13. Yes; T PTQ T RTS by the SAS Congruence Postulate because PT **** RT **** (Given), a PTQ a RTS (Vertical Angles Theorem), and ST **** QT **** (Given). 14. Yes; T PQR T PSR by the SSS Congruence Postulate 15. a S a X 16. **** AB FG **** 18. Statements 19. 17. a JKM a LMK 2. a JMN a JKL 2. All right angles are . 3. a J a J 3. Reflexive Prop. of Cong. 4. MN **& KL **** 5. T JMN T JKL 4. Given 6. JM **** JK **** 6. Corresponding parts of triangles are . Reasons 1. Given 2. a Q a S 2. All right angles are . 3. a QRP a SRT 3. Vertical Angles Theorem 4. **** PR TR *& 5. T QRP T SRT 4. Given 6. QR **** SR **** 6. Corresponding parts of triangles are . 26. QR QP 2x 1 x 4 1. Given 2x 1 x x 4 x 2. T UZV and T XYW are right triangles. 2. Definition of right triangle 3. UV XW **** **& 4. UZ **** XY **** 5. T UZV T XYW 3. Given QR 2x 1 2(5) 1 9 4. Given 5. HL Congruence Theorem A B L M J J K N AAS Congruence Theorem 21. P S R R T AAS Congruence Theorem 22. Statements XY ZY 3x 4 6x 1 3x 4 3x 6x 1 3x 4 1 3x 1 1 3 3x 3 3x 3 3 1x XY 3x 4 3(1) 4 7 28. Yes; all three properties of a reflection are met; one line Reasons 1. a C and a D are right angles. 1. Given 2. T ABC and T BAD are right triangles. 2. Definition of right triangle 3. **** AC BD **** 4. **** AB **** AB 3. Given of symmetry. 29. No; the orientation is not reversed; one line of symmetry. 30. Yes; all three properties of a reflection are met; no lines of symmetry. 31. Sample answer: 4. Reflexive Prop. of Cong. 5. T ABC T BAD 5. HL Congruence Theorem 6. a CBA a DAB 6. Corresponding parts of triangles are . Copyright © McDougal Littell Inc. All rights reserved. 27. 4 3x 1 HL Congruence Theorem 20. x14 x1141 x5 D B 5. AAS Congruence Theorem 25. JM LM 3 1. a UZV and a XYW are right angles. A 5. AAS Congruence Theorem 1. a Q and a S are right angles. Reasons C 1. Given 24. Statements 12. Yes; T DEF T ACB by the SAS Congruence Postulate because PQ **** PS **** (Given), QR **** SR **** (Given), and PR **** PR (Reflexive Property of Congruence). **** Reasons Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 83 Chapter 5 continued Chapter 5 Chapter Test (p. 296) 12. The figure has two 13. The figure has four lines of symmetry. 1. RS **** XY **** , ST **** YZ **** , TR *& ZX **** , lines of symmetry. a R a X, a S a Y, a T a Z; T RST T XYZ, T STR T YZX, T TRS T ZXY, T TSR T ZYX, T SRT T YXZ, or T RTS T XZY 2. ma L ma U 46 3. TU KL 10 4. SSS Congruence Postulate; BC **** BC **** (Reflexive Property of Congruence), so all three sides of T ABC are congruent to all three sides of T DBC. Chapter 5 Standardized Test (p. 297) 1. C 2. J 3. B AD CD 4. F; 4x 3 2x 3 5. HL Congruence Theorem; by the Reflexive Property of Congruence, HF **** HF ****, so the hypotenuse and leg of right T HFE are congruent to the hypotenuse and corresponding leg of T HFG. 4x 3 2x 2x 3 2x 2x 3 3 2x 3 3 3 3 6. Yes; vertical angles are congruent, so the triangles are 2x 6 2x 6 2 2 x3 congruent by the AAS Congruence Theorem. 7. Yes; vertical angles and alternate interior angles are congruent, so the triangles are congruent by the AAS Congruence Theorem. Reasons 5. C; AD 4x 3 4(3) 3 12 3 9 1. BC **** DC **** 1. Given 6. J; 2. a B and a D are right angles. 2. Given 3. a B a D 3. Right angles are . 4. a ACB a ECD 4. Vertical Angles Theorem 5. T ABC T EDC 5. ASA Cong. Postulate AB ED 6. **** **** 6. Corresponding parts of triangles are . 8. Statements 7. y B A 1 9. PR PQ C 1 x Cⴕ 2x 8 2x 8 2 2 x4 8. A is (2, 2); B is (3, 4); C is (5, 1). PR 2x 2(4) 8 9. The x-coordinate of each vertex in the original triangle is Aⴕ Bⴕ the same as the x-coordinate of the vertex in the reflected triangle. The y-coordinate of each vertex in the original triangle is the opposite of the y-coordinate of the vertex in the reflected triangle. ST UT 10. 5x 2x 3 5x 2x 2x 3 2x 3x 3 3x 3 3 3 x1 ST 5x 5(1) 5 11. No; the orientation of the figure is not reversed. Chapter 5 Algebra Review (p. 299) 1. 3x 2y 13 Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 5x 4y 4 3(3) 2(2) ⱨ 13 5(0) 4(1) ⱨ 4 9 4 ⱨ 13 0 (4) ⱨ 4 13 13 44 (3, 2) lies on the line. 84 2. (0, 1) lies on the line. Copyright © McDougal Littell Inc. All rights reserved. Chapter 5 continued 3. y 3x 8 4. 4 ⱨ 3(4) 8 x 6y 2 (4) 6(1) ⱨ 2 4 ⱨ 12 8 4 6 ⱨ 2 4 20 2 2 (4, 4) does not lie on the line. 5. x 7y 5 (4, 1) does not lie on the line. 2x 4y 2 6. (5) 7(0) ⱨ 5 2(1) 4(1) ⱨ 2 5 0 ⱨ 5 2 4 ⱨ 2 5 5 2 2 (5, 0) does not lie on the line. (1, 1) does not lie on the line. y2 y1 71 6 7. slope x2 x1 83 5 y2 y1 5 4 9 8. slope not defined x2 x1 33 0 y2 y1 7 1 (6) 9. slope 1 x2 x1 7 1 6 y2 y1 7 81 10. slope x2 x1 3 2 (5) y2 y1 44 0 11. slope 0 x2 x1 4 (1) 5 y2 y1 6 (4) 10 5 12. slope x2 x1 2 (2) 4 2 Copyright © McDougal Littell Inc. All rights reserved. Geometry, Concepts and Skills Chapter 5 Worked-Out Solution Key 85