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Transcript
4.2 Apply Congruence and Triangles Hubarth Geometry Corresponding Parts- are angles and sides that are in the same position of their respected shape. Congruent- same size and shape Congruent Same size and shape Not Congruent Different sizes and shapes Congruence Statements When you write a congruence statement for two polygons, always list the corresponding vertices in the same order. You can write congruence statements in more than one way. Two possible congruence statements for the triangles at the right. ABC FDE or BCA DEF CorrepondingAngles : A F , B D, C E Correponding Sides : AB FD, BC DE , AC FE F A B C D E Ex 1 Identifying Congruent Parts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. JKL TSR J T , K S , L R JK TS , KL SR, LJ RT Ex 2 Use Properties of Congruent Figures In the diagram, DEFG SPQR. a. Find the value of x. b. Find the value of y. a. You know that FG FG = QR QR. b. You know that m F =m F Q. Q 12 = 2x – 4 68 = (6y + x) 16 = 2x 68 = 6y + 8 8 =x 10 = y Ex 3 Show that Figures are Congruent 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. 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 . 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 All corresponding parts are congruent, so AJKD BC . By the Reflexive Property, JK CKJB. KJ E Theorem Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. B A F D C If A D and B E, then C F EX 4 Use the Third Angles Theorem Find mBDC A B and ADC BCD, so by the Third Angles Theorem, ACD BDC. By the Triangle Sum Theorem, mACD 180 - 45 - 30 105 So, mACD mBDC 105 Ex 5 Prove that Triangles are Congruent Write a proof. Given : AD CB, DC AB, ACB CAB, CAD ACB Prove : ACD CAB Plan for Proof a. Use the Reflexive Property to show AC AC b. Use the Third Angles Theorem to show that B D. STATEMENTS REASONS 1. AD CB, DC BA 1. Given 2. AC AC 2. Reflexive Property of Congruence 3. ACD CAB CAD ACB 3. Given 4. B D 4. Third Angles Theorem 5. ACD CAB 5. Def. of Practice In the diagram at the right, ABGH CDEF. 1. Identify all pairs of congruent corresponding parts. AB CD, AH FC , GH EF , BG DE A C , H F , G E , B D 2. Find the value of x and find m H. H 105 You know that H F (4x+ 5)° = 105° 4x = 100 x = 25 3. Show that PTS RTQ All of the corresponding parts of PTS are congruent to those of RTQ by the indicated markings, the Vertical Angle Theorem and the Alternat Interior Angle Theorem. 4. In the diagram, what is mDCN. DCN 75 5. By the definition of congruence, what additional information is needed to know that DCN SRN DC RS and DN SN