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LESSON Chapter Review 4 4-1 Classifying Triangles W Classify each triangle by its angle measure. Z 1. nXYZ 2. nXYW 3. nXZW 25 35 X Classify each triangle by its side lengths. 4. nDEF 60 2 5. nDEG Y D G 6. nEFG 8 10 E F 4-2 Angle Relationships in Triangles Find each angle measure. D 7. m ACB 4x – 45 A 8. m K K C 2x – 3 x–7 M B 25° 3x 2x N L 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 32.5° and 47.5°. Find the measure of the third angle. 4-3 Congruent Triangles Given nABC ù nXYZ. Identify the congruent corresponding parts. 10. BC ù 11. ZX ù 12. 13. Aù L Given nJKL ù nPQR. Find each value. 14. x Yù 42° 15. RP J Copyright © by Holt, Rinehart and Winston. All rights reserved. 93 R 3y – 2 K 2y – 1 22 3x° Q P Geometry CHAPTER 4 REVIEW CONTINUED 16. Given: < zz k; BD ù CD ; AB ù AC ; AD ⊥ CB ; AD ⊥ XW ; XAC ù WAB x Prove: nABD ù nACD k y z C Statements D 1. 2. AD ù AD 2. 3. < zz k; AD ⊥ CB ; AD ⊥ XW 3. 4. 4. Def. of ⊥ lines ADB ù 5. ADC 6. Given 6. 7. XAC ù ACD; WAB ù ABD 7. 8. Transitive Property of Congruence 8. 9. B Reasons 1. BD ù CD ; AB ù AC ; 5. w A CAD ù 9. BAD 10. 10. Def of Congruent Triangles 4-4 Triangle Congruence: SSS and SAS 17. Given that HIJK is a rhombus, use SSS to explain why nHIL ù nJKL. H L K Copyright © by Holt, Rinehart and Winston. All rights reserved. 94 I J Geometry CHAPTER 4 REVIEW CONTINUED 18. Given: NR ù PR ; MR ù QR M N Prove: nMNR ù nQPR R P Q 4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 19. nHIK ù nJIK 20. nPQR ù nRSP H K P Q I S J 21. Use ASA to prove the triangles congruent. R C A B Given: BD bisects ABC and ADC Prove: nABD ù nCBD D Copyright © by Holt, Rinehart and Winston. All rights reserved. 95 Geometry CHAPTER 4 REVIEW CONTINUED 4-6 Triangle Congruence: CPCTC X U Z 22. Given: UZ ù YZ , VZ ù XZ Y Prove: XY ù VU V 4-7 Introduction to Coordinate Proof Position each figure in the coordinate plane. 23. a right triangle with legs 3 and 4 units in length 24. a rectangle with sides 6 and 8 units in length y y 4 8 6 4 2 2 x 2 Copyright © by Holt, Rinehart and Winston. All rights reserved. 4 x 2 4 6 8 96 Geometry CHAPTER 4 REVIEW CONTINUED 4-8 Isosceles and Equilateral Triangles 25. Assign coordinates to each vertex and write a coordinate proof. y Given: rectangle ABCD with diagonals intersecting at z Prove: CZ ù DZ x Find each angle measure. A 3x 26. m B C x 27. m HEF E B 110° H F G 28. Given: nPQR has coordinates P(0, 0), Q(2a, 0), and R(a, aÏ3 w) Prove: nPQR is equilateral. y 16 12 8 4 x P Copyright © by Holt, Rinehart and Winston. All rights reserved. 97 4 8 12 16 Q Geometry CHAPTER Postulates and Theorems 4 Theorem 4-2-1 Corollary 4-2-2 Corollary 4-2-3 (Triangle Sum Theorem) The sum of the angle measures of a triangle is 180°. m A m B m C ! 180° The acute angles of a right triangle are complementary. The measure of each angle of an equilateral triangle is 60°. Theorem 4-2-4 (Exterior Angle Theorem) The measure on an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Theorem 4-2-5 (Third Angles Theorem) If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. (Side-Side-Side (SSS) Congruence) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Postulate 4-4-1 Postulate 4-4-2 Postulate 4-5-1 Theorem 4-5-2 Theorem 4-5-3 Theorem 4-8-1 Converse 4-8-2 Corollary 4-8-3 Corollary 4-8-4 (Side-Angle-Side (SAS) Congruence) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. (Angle-Side-Angle (ASA) Congruence) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (Angle-Angle-Side (AAS) Congruence) If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of another triangle, then the triangles are congruent. (Hypotenuse-Leg (HL) Congruence) If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. (Isosceles Triangle Theorem) If two sides of a triangle are congruent, then the angles opposite the sides are congruent. (Converse of Isosceles Triangle Theorem) If two sides of a triangle are congruent, then the angles opposite the sides are congruent. (Equilateral Triangle) If a triangle is equilateral, then it is equiangular. (Equiangular Triangle) If a triangle is equiangular, then it is equilateral. Copyright © by Holt, Rinehart and Winston. All rights reserved. 98 Geometry