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Geometry Section 6-2A Proofs with Parallelograms 1 Proofs with Parallelograms: We have been working on developing skills in writing proofs. Each proof has become increasingly difficult and you have been asked to fill in more and more as time has gone by. You must continue to build this skill so that you can write a proof from scratch all by yourself. 2 Proofs: 5 steps to writing a proof. 1. 2. 3. 4. Rewrite Draw State (“Given” and “Prove”) Plan a. Think backwards. b. Do you need to prove things about congruent angles, parallel lines, triangles, etc? 5. Demonstrate (Write the proof) 3 We have not spent as much time on the planning steps as we have on the other steps. Today we will focus on that as well as writing a proof from scratch. We will be focusing on parallelograms because they have many properties that you know well. a. mPMN 135o b. mMNO 45o c. mOPM 45o d. MP 7 e. OP 15 f. MQ 5.5 g. NQ 10.5 4 Writing a Proof Prove: The opposite angles of a parallelogram are congruent. Rewrite: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Draw: A B D C State: Given: ABCD is a parallelogram Prove: ABC @ CDA, DAB @ BCD Plan: If we can divide this into 2 triangles and prove that they are congruent, then we can use CPCTC to match up congruent angles. How do we divide this into 2 triangles? Draw an auxiliary line. 5 B A Given: ABCD is a parallelogram Prove: ABC @ CDA, BAD @ BCD D ABCD is a parallelogram Draw AC AB DC Given Two pts. determine a line Def. of parallelogram AD BC ACD @ CAB DAC @ BCA AC @ AC DABC @ DCDA ABC @ CDA Draw BD ABD @ BDC ADB @ DBC Def. of parallelogram Alt. Int. ‘s are @. Alt. Int. ‘s are @. Reflexive Property ASA CPCTC Two pts. determine a line Alt. Int. ‘s are @. Alt. Int. ‘s are @. BD @ BD DBAD @ DBCD BAD @ BCD Reflexive Property ASA CPCTC C 6 Properties of Parallelograms: 1. Opposite sides of a parallelogram are parallel. 2. Opposite angles of a parallelogram are congruent. 3. Opposite sides of a parallelogram are congruent. 4. Consecutive angles of a parallelogram are supplementary. 5. Diagonals of a parallelogram bisect each other. 7 If I give you 3 dots on a coordinate grid, how many different parallelograms could we make? 8 Homework: Practice 6-2A Change #12 to Prove: AB @ CD and BC @ AD 9