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4.2 Apply Congruence and Triangles • You will identify congruent figures. • Essential Question: What are congruent figures? Tell students they will learn how to answer this question by studying the definition of congruent figures Warm-Up1Exercises EXAMPLE Identify congruent parts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. SOLUTION The diagram indicates that Corresponding angles J Corresponding sides JK JKL T, K TS, KL TSR. S, SR, LJ L R RT 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. Warm-Up2Exercises EXAMPLE Use properties of congruent figures b. You know that F m F=m Q 68 o = (6y + x) 68 = 6y + 8 10 = y Q. o 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-Up3Exercises EXAMPLE Show that figures are congruent Then, 1 4 and 2 3 by the Alternate Interior Angles Theorem. So, all pairs of corresponding angles are congruent. The diagram shows AJ CK , KD JB , and DA BC . By the Reflexive Property, JK KJ . All corresponding parts are congruent, so AJKD CKJB. Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, 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) You know that (4x+ 5)° = 105° 4x = 100 x = 25 (b) You know that H m H m F =105° H F F CDEF. H. Warm-Up Exercises GUIDED PRACTICE for Examples 1, 2, and 3 In the diagram at the right, ABGH 3. Show that PTS CDEF. RTQ. SOLUTION All of the corresponding parts of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternate Interior Angle theorem. 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 AD CB, DC ACD PROVE AB CAB, ACD CAD ACB 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° Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 5. By the definition of congruence, what additional information is needed to know that NDC NSR. ANSWER DC RS and DN SN Daily Homework Quiz Warm-Up Exercises In the diagram, 1. m A = ? ANSWER 2. FD 60° ? ANSWER 3. EDF ANSWER ABC CA ? BAC DEF. Complete the statement. Daily Homework Quiz Warm-Up Exercises 4. Write a congruence statement for the two small triangles. Explain your reasoning. ANSWER WXZ YXZ; The diagram tell us that W Y and WZX YZX. WXZ YXZ by the Third s Thm. From the diagram WX YX and WZ YZ, and XZ XZ by Refl. Prop. Of Segs. • You will identify congruent figures. • Triangles can be proved congruent by showing that all 3 pairs of corresponding sides and all 3 pairs of corresponding angles are congruent. • If two angles of one triangle are congruent to two angles of another, then the third angles are congruent. • Essential Question: What are congruent figures? Congruent figures are figures that have exactly the same size and shape.