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Warm-Up Exercises 1. When are two angles congruent? ANSWER when they have the same measure 2. In ∆ABC, if m A = 64º and m ANSWER B = 71º, what is m C? 45º 3. What property of angle congruence is illustrated by this statement? If A B and B C, then A C. ANSWER Transitive Property Warm-Up Exercises Target Proving triangles congruent. GOAL: 4.2 Identify congruent figures. Warm-Up Exercises Vocabulary • congruent figures - figures the same shape and size; two figures are congruent if and only if all corresponding angles and all corresponding sides are congruent. CPCTC - Corresponding Parts of Congruent Triangles are Congruent Third Angles Theorem 4.3 – If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. R-S-T Theorem for Congruent Triangles 4.4 – Congruence of triangles is reflexive, symmetric, and transitive. Warm-Up1Exercises EXAMPLE Identify congruent parts To write a congruence statement for the triangles first identify all pairs of congruent corresponding parts. SOLUTION Corresponding angles Corresponding sides J JK The diagram indicates that T, K TS, KL JKL S, SR, LJ L RT TSR. congruence statement R Warm-Up2Exercises EXAMPLE Use properties of congruent figures In the diagram, DEFG SPQR. a. Find the value of x. b. Find the value of y. SOLUTION a. You know that FG FG = QR 12 = 2x – 4 16 = 2x 8=x QR. b. You know that F m F=m Q 68 o = (6y + x) 68 = 6y + 8 10 = y o Q. Warm-Up3Exercises EXAMPLE Show that figures are congruent PAINTING If you divide the wall into orange and blue sections along JK , will the sections of the wall be the same size and shape?Explain. SOLUTION From the diagram, A C and D B because all right angles are congruent. Also, by the Lines Perpendicular to a Transversal Theorem, AB DC . Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the below, ABGH CDEF. 1. Identify all pairs of congruent corresponding parts. SOLUTION Corresponding sides: Corresponding angles: AB CD, BG DE, GH FE, HA FC A G C, B E, H D, F. Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH 2. Find the value of x and find m SOLUTION (a) (b) You know that H (4x+ 5)° = 105° 4x = 100 x = 25 You know that H m H m F =105° F F CDEF. H. Warm-Up4Exercises EXAMPLE Use the Third Angles Theorem Find m BDC. SOLUTION A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, m ACD = 180° – 45° – 30° = 105° . ANSWER So, m ACD = m BDC = 105° by the definition of congruent angles. Warm-Up5Exercises EXAMPLE Prove that triangles are congruent Write a proof. GIVEN PROVE AD CB, DC AB ACD CAB, CAD ACB ACD CAB Plan for Proof a. Use the Reflexive Property to show that AC AC. b. Use the Third Angles Theorem to show that B D Warm-Up5Exercises EXAMPLE Prove that triangles are congruent Plan in Action STATEMENTS REASONS 1. AD CB, DC a. 2. AC AC. BA 1. Given 2. Reflexive Property of Congruence 3. b. 4. 5. ACD CAD B ACD CAB, ACB D 3. Given 4. Third Angles Theorem CAB 5. Definition of Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 4. In the diagram, what is m DCN. SOLUTION CDN NSR, DNC SNR then the third angles are also congruent NRS DCN = 75° 5. By the definition of congruence, what additional information is needed to know that NDC NSR. ANSWER DC RS and DN SN