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Math 53 "Winter ’09" 3.2 "Corresponding Parts of Congruent Triangles" ————————————————————————————————————————————————— Objectives: * Using CPCTC to establish congruency in triangles * How to plan a proof * Properties of right triangles ————————————————————————————————————————————————— Key Concepts: CPCTC Hypotenuse and Legs of a Right Triangle HL Pythagorean Theorem Square Roots Property ————————————————————————————————————————————————— Preliminaries: Recall that the de…nition of congruent triangles states that all six parts of one triangle are congruent respectively to the six corresponding parts of the second triangle. Example 1: (Congruent triangles) Given: R 6 6 Prove: and 6 V are right 6 s 1=6 2 RST = V ST CPCTC: kCorresponding parts of congruent triangles are congruentk Page: 1 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein Note: 3.2 For triangles that have been proved congruent, CPCTC may be used to establish that either two lines segments or two angles are congruent. Example 2: (Using CPCTC) Given: Prove: HJ ? KL and HK = HL KJ = JL Example 3: (Planning a proof) Given: Prove: ZW = Y X and ZY = W X ZY W X Because many proofs depend on establishing congruent triangles, we need to consider the following suggestions Suggestions for a proof that involves congruent triangles 1: Mark the …gures systematically, using the same number of dashes on congruent sides, the same number of arcs on congruent angles, and a square in each right angle. 2: Trace the triangles to be proved congruent in di¤erent colors. 3: If the triangles overlap, draw them separately. Page: 2 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 3.2 Theorem 3.2.1: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent (HL) Example 4: (Using Theorem 3.2.1) Given: 6 1 and 6 2 are right 6 s H is the midpoint of F K F G HJ Prove: F G = HJ Pythagorean Theorem: The square of the length of the hypotenuse of a right triangle equals the sum of squares of the lengths of the legs of the right triangle Square Roots Property Let x represent the length of a line segment, and let p represent a positive number. If x2 = p; then : Page: 3 Bibiana Lopez Elementary Geometry by Alexander and Koeberlein 3.2 Example 5: (Using the Pythagorean Theorem) Given: Find: ABC is a right triangle, a = 4; and b=3 c Example 6: Given: DB ? BC and CE ? DE AB = AE Prove: BDC = ECD Page: 4 Bibiana Lopez