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Geometry Lesson 4.2 Introduction to Congruent Triangles Warm-Up: Review of “Congruent” Congruent segments have the same length _______________ Congruent angles have the same measure _______________ 1. Definition of Congruent Figures Geometric figures are congruent if they have exactly the same size and exactly the same shape Congruent NOT congruent Congruence Implies Correspondence Congruent figures have corresponding angles and corresponding sides that are congruent B A ABC DEF C A D Angles B E C F E D AB DE BC EF AC DF F Sides Get the Order Right! CORRESPONDING PARTS MUST MATCH! B A E C D Can also write BCA EFD, but can NOT write ABC EFD F Example 1: Naming Congruent Parts Write a congruence statement and name all corresponding angles and sides Q Congruence Statement QRP ACB R P C A Q A QR AC R C PR BC P B PQ AB B Practice 1: Naming Congruent Parts Write a congruence statement and name all corresponding angles and sides _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ Example 2: Finding Measures In the figures below, NPLM EFGH Find the value of x and y 8 L P F M 110° G (2x-3) 72° 10 N E (7y+9)° H 1. What angle corresponds to E? N 7y + 9 = 72 y=9 2. What side corresponds to GH? LM 2x – 3 = 8 x = 5.5 Practice 2: Finding Measures In the figures, ABCDEF GHIJKL Find values for x and y 2. Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent B A E 3rd C D If 2s , then 3rd s 3rd F Example 3: Third Angles Theorem Find the value of x Start by writing a congruence statement: ABC DEF Then, apply the third angles theorem: C F mC = mF From ABC, mC = 40° ( sum theorem) mD = x° = 40° Practice 3: Third Angles Theorem Find x & mF using the third angles theorem B E 65° A 55° C D F 3. Proving Triangles are Congruent Process: Use the definition of congruence and the properties of the figure to develop a logical argument Example 4: Given the diagram, prove PQR NQM N R 92° 92° P Q M Example 4, cont. N R 92° 92° Q P 1. What do you need to know? That all corresponding sides and angles are 2. What do you already know? All sides are and mP = mN 3. What can you show? PQR NQM (vertical s) and R M (3rd s thm) M Example 4, cont. N R 92° 92° Statement P Q Reason RP MN, RQ MQ, PQ NQ Given mP = mN Given P N PQR MQN R M PQR NQM M If =, then If vertical s, then If 2s , then 3rd s If corresp. s and sides , then figures 4. Congruent Triangle Properties Reflexive Every triangle is congruent to itself Symmetric If ABC DEF, then DEF ABC Transitive If ABC DEF and DEF JKL, then ABC JKL B A E C D K F J L Closure The design has only congruent triangles If the total area is 96 ft², what is the area of one triangle 32 congruent triangles 96 ft² / 32 = 3 ft² per triangle Assignment Ch 4.2 w/s