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4.1 Congruent Figures (Page 198- 203) Congruent figures- two or more figures with the same size and shape There are three ways to show congruency by using overlapping figures. Congruent polygons- all their corresponding parts are congruent Congruence statement- tells that two figures are congruent by naming their lettered angles. For example, △ RBW @ △HJG Theorem 4-1 (AKA the “Triangle Third Angle Congruency Theorem”)If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent. This theorem can be applied in a proof like this: Given: PQ ≅ PS, QR ≅ SR, ÐQ ≅ ÐS, ÐQPR ≅ ÐSPR Prove: △PQR ≅ △PSR Steps 1. PQ ≅ PS, QR ≅ SR Reason 1. Given Explanation 1. We only need this much of the given right now, we’ll use the rest later 2. PR ≅ PR 2. Reflexive property of Congruence 2. PR is congruent to itself. 3. ÐQ ≅ ÐS, ÐQPR ≅ ÐSPR 3. Given 3. Here we use the rest of the given 4. Triangle Third Angle Congruency Theorem 4. We already know that the other angles are congruent so we can say that those two are congruent. 5. Definition of Congruent Triangles 5. Think back to the given. Now we have all 6 parts of both triangles, so we can prove them both congruent. 4. ÐQRP ≅ ÐSRP 5. △PQR ≅ △PSR Here are some other problems using congruent figures: Can you conclude the figures are congruent? Explain why. Yes. TK ≅ TK because of the “reflexive property of congruence.” TKU ≅ TKR because of the “triangle third angle congruency theorem.” Therefore, all sides and angles are congruent so the triangles are congruent by the definition of congruent triangles. Write a congruence statement for the pair of triangles. △JYB @ △XCH (this is a problem where it’s imperative to remember to put the letters in the right order) Now try on your own: ****Before you start the proof, here are some reminders: 1. When naming two figures, put letters in the same order! 2. In this section you haven’t learned AAS, SSS, and so on, so don’t use them to solve the proof! 1. Easy △WYS @ △MKV. List the congruent corresponding parts. Use three letters (in the right order!) to name each angle. 2. Medium Can you conclude that the figures are congruent? Justify your answer. 3. Hard PROOF Given: ÐA @ ÐE, ED @ AB , EC CD AF , FB @ CD Prove: △AFB @ △ECD ANSWER KEY 1. Question: △WYS @ △MKV. List the congruent corresponding parts. Use three letters (in the right order!) to name each angle. Answer: ÐW @ ÐY, ÐS @ ÐM, ÐK @ ÐV, WY @ MK , YS @ KV , SW @ VM 2. Question: Can you conclude that the figures are congruent? Justify your answer. Answer: Yes. All three sides are congruent (CA congruent to AC because of reflexive). Angle ACD is congruent to Angle CAB because AB and DC are parallel so they are vertical angles. 3. Question: Given: ÐA @ ÐE, ED @ AB , EC CD AF , FB @ CD Prove: △AFB @ △ECD Answer: 1. ÐA @ ÐE, ED @ AB , EC CD AF , 1. Given FB @ CD 2. ÐC @ ÐF 3. ÐD @ ÐB 4. △AFB @ △ECD 2. Right angle congruency theorem 3. Third angle congruency theorem 4. Definition of congruent triangles