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Chapter 5 Summary and Review Additional Resources The following resources are available to help review the materials in this chapter. VOCABULARY • corresponding parts, p. 233 • perpendicular bisector, p. 274 • congruent figures, p. 233 • distance from a point to a line, p. 273 • proof, p. 243 • equidistant, p. 273 • line of symmetry, p. 284 Chapter 5 Resource Book • Chapter Review Games and Activities, p. 75 • Cumulative Review, Chs. 1–5 • reflection, p. 282 VOCABULARY REVIEW Fill in the blank. 1. When two figures are __?__, their corresponding sides and their corresponding angles are congruent. congruent 2. A(n) __?__ is a convincing argument that shows why a statement is true. proof 3. If a point is on the __?__ of a segment, then it is equidistant from the endpoints of the segment. perpendicular bisector 4. If a point is on the angle bisector of an angle, then it is __?__ from the two sides of the angle. equidistant 5. A(n) __?__ is a transformation that creates a mirror image. reflection 5.1 Examples on pp. 233–235 CONGRUENCE AND TRIANGLES In the diagram, TABC c TRST. Identify all corresponding congruent parts. B Corresponding angles Corresponding sides A EXAMPLE aA c aR aB c aS aC c aT &* c RS &* AB &* c RT &* AC &* c ST &* BC T R C Use the triangles shown at the right to determine whether the given angles or sides represent corresponding angles, corresponding sides, or neither. corresponding angles &* and QR &* 6. aJ and aR neither 7. KL 8. aK and aQ corresponding sides &* and LJ & neither 10. JK &* and PR &* neither 11. aR and aL L 9. PQ corresponding angles S J K R P P Chapter Summary and Review 291 291 Chapter Summary and Review continued 5.2 Examples on pp. 241–244 PROVING TRIANGLES ARE CONGRUENT: SSS AND SAS EXAMPLES a. Tell which congruence postulate you would use to show that the triangles are congruent. L b. A B G C H P M N F a. In the triangles shown, b. In the triangles shown, &* c FG &*, S AB &* c GH &**, and S BC &* c FH &**. S AC So, TABC c TFGH by the SSS Congruence Postulate. &** c NP &*, S LM A aMLN c aPNL, and &** c NL &**. S LN So, TLMN c TNPL by the SAS Congruence Postulate. Decide whether enough information is given to show that the triangles are congruent. If so, tell which congruence postulate you would use. 12. E B 13. A P F D C yes; SAS Congruence Postulate 5.3 a. R P S yes; SSS Congruence Postulate P yes; SAS Congruence Postulate Examples on pp. 250–253 Tell which congruence postulate or theorem you would use to show that the triangles are congruent. D E F K L A aE c aK, &* c KL &*, and S EF A aF c aL. So, TDEF c TJKL by the ASA Congruence Postulate. Chapter 5 S b. J a. In the triangles shown, 292 R T PROVING TRIANGLES ARE CONGRUENT: ASA AND AAS EXAMPLES 292 14. P S Congruent Triangles U T Y Z X b. In the triangles shown, A aU c aZ, A aT c aY, and &* c XY &*. S ST So, TSTU c TXYZ by the AAS Congruence Theorem. Chapter Summary and Review continued Determine what information is needed to use the indicated postulate or theorem to show that the triangles are congruent. 15. AAS Congruence 16. ASA Congruence Theorem T 17. ASA Congruence Postulate Postulate B U X A K L C J G S Z Y aS c aX 5.4 H M F &* c FG &* AB aJKM c aLMK Examples on pp. 257–259 HYPOTENUSE-LEG CONGRUENCE THEOREM: HL Prove that TABC c TBFD. EXAMPLE &* ∏ AF &*, DF &** ∏ AF &* Given CB &*. B is the midpoint of AF &* c BD &* AC Prove TABC c TBFD A B C F D Show that the triangles are right triangles, the hypotenuses are congruent, and that corresponding legs are congruent. Statements Reasons &* ∏ AF &*, DF &** ∏ AF &*, and B is 1. CB 1. Given &*. the midpoint of AF 2. aCBA and aDFB are right angles. 2. ∏ lines form right angles. 3. TABC and TBFD are right triangles. 3. Definition of right triangle &* c BD &* H 4. AC 4. Given &* c BF &* L 5. AB 5. Definition of midpoint 6. TABC c TBFD 6. HL Congruence Theorem 18. Use the information given in the diagram to fill in the missing V W Z Y statements and reasons to prove that TUZV c T XYW. U Statements Reasons 1. aUZV and aXYW are right angles. Given 1. _________?_________ 2. TUZV and TXYW are right 2. _________?_________ X Def. of right T triangles. UV &* c XW &* 3. _________?_________ 3. Given &** c XY &* 4. UZ Given 4. _________?_________ a UZV c a XYW 5. _________?_________ 5. HL Congruence Theorem Chapter Summary and Review 293 293 Chapter Summary and Review continued 19–21. Check sketches. 19. HL Congruence Theorem 20. AAS Congruence Theorem 21. AAS Congruence Theorem 22. Statements (Reasons) 1. aC and aD are right angles. (Given) 2. TABC and TBAD are right triangles. (Definition of right triangle) & c BD & (Given) 3. AC & c AB & (Reflexive Prop. of 4. AB Congruence) 5. TABC c TBAD (HL Congruence Theorem) 6. aCBA c aDAB (Corresp. parts of c triangles are c.) 23. Statements (Reasons) 1. aJMN and aJKL are right angles. (Given) 2. aJMN c aJKL (All right angles are congruent.) & c KL & (Given) 3. MN 4. aJ c aJ (Reflexive Prop. of Congruence) 5. TJMN c TJKL (AAS Congruence Theorem) & c JK & (Corresp. parts of 6. JM c triangles are c.) 24. Statements (Reasons) 1. aPQR and aTSR are right angles. (Given) 2. aPQR c aTSR (All right angles are congruent.) & c TR & (Given) 3. PR 4. aPRQ c aTRS (Vertical Angles Theorem) 5. TPQR c TTSR (AAS Congruence Theorem) & c SR & (Corresp. parts of 6. QR c triangles are c.) 5.5 Examples on pp. 265–267 USING CONGRUENT TRIANGLES EXAMPLE &** c KN &* and aKML and aKNJ In the diagram, KM &* c KJ &*. are right angles. Prove that KL L K M J First show that TMKL c TNKJ. Then use the fact that corresponding &* c KJ &. parts of congruent triangles are congruent to show that KL L Sketch the triangles separately. Mark the given information and any other information you can conclude from the diagram. N K M K J Statements Reasons &** c KN &* 1. KM 1. Given 2. aKML and aKNJ are right angles. 2. Given 3. aKML c aKNJ 3. Right angles are congruent. 4. aK c aK 4. Reflexive Prop. of Congruence 5. TMKL c TNKJ 5. ASA Congruence Postulate &* c KJ &* 6. KL 6. Corresponding parts of congruent triangles are congruent. Sketch the overlapping triangles separately. Use the given information to mark all congruences. Then tell what theorem or postulate you can use to show that the triangles are congruent. 19–24. See margin. &* c BD &* 19. AC C && c KL &* 20. MN D &* c TR &* 21. PR M L J A B P P K N 22. Use the diagram and the information given in Exercise 19 above to prove that aCBA c aDAB. 23. Use the diagram and the information given in Exercise 20 above &* c JK &*. to prove that JM 24. Use the diagram and the information given in Exercise 21 above &* c SR &*. to prove that QR 294 294 N Chapter 5 Congruent Triangles R S T Chapter Summary and Review continued 5.6 Examples on pp. 273–275 ANGLE BISECTORS AND PERPENDICULAR BISECTORS EXAMPLE Find AB. C By the Angle Bisector Theorem, AB DB. A 2x 1 3x 2 Use Angle Bisector Theorem. 2x 3 3x Add 2 to each side. 3x 31. Sample answer: D 2x 1 3x 2 B Subtract 2x from each side. You are asked to find AB, not just the value of x. ANSWER AB 2x 1 2(3) 1 7 Use the diagram to find the indicated measure. 25. Find JM. 3 26. Find QR. 9 27. Find XY. 7 3 J x4 3x 4 2x 1 W L P K 5.7 X P M S Z Examples on pp. 282–285 REFLECTIONS AND SYMMETRY EXAMPLE Y 6x 1 R Tell whether the red figure is a reflection of the blue figure. Then determine the number of lines of symmetry in the blue figure. k The red figure is a reflection of the blue figure in line k. The blue figure has one line of symmetry. Tell whether the red figure is a reflection of the blue figure. Then determine the number of lines of symmetry in the blue figure. 28. yes; 1 29. k no; 1 k 30. k yes; 1 31. Draw a six-sided figure that has two lines of symmetry. See margin. Chapter Summary and Review 295 295