1 Geometry Section 6-2A Proofs with Parallelograms

2 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.

3 5 steps to writing a proof. 1. Rewrite Proofs: 2. Draw 3. State (“Given” and “Prove”) 4. Plan a. Think backwards. b. Do you need to prove things about congruent angles, parallel lines, triangles, etc? 5. Demonstrate (Write the proof)

4 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. m  PMN 135 o b. m  MNO 45 o c. m  OPM 45 o d. MP 7 e. OP 15 f. MQ 5.5 g. NQ 10.5

5 Writing a Proof Prove: The opposite angles of a parallelogram are congruent. Rewrite: If then a quadrilateral is a parallelogram, its opposite angles are congruent. Draw: AB C D State:Given: Prove: ABCD is a parallelogram  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.

6 Draw ACTwo pts. determine a line ABCD is a parallelogram Given AB  DC Def. of parallelogram AD  BC Def. of parallelogram A B CD  ACD   CABAlt. Int.  ‘s are .  DAC   BCAAlt. Int.  ‘s are . AC  AC Reflexive Property  ABC   CDA ASA  ABC   CDA CPCTC Given: Prove: ABCD is a parallelogram  ABC   CDA,  BAD   BCD  ABD   BDCAlt. Int.  ‘s are .  ADB   DBCAlt. Int.  ‘s are . BD  BD Reflexive Property  BAD   BCD ASA  BAD   BCD CPCTC Draw BD Two pts. determine a line

7 1. Opposite sides of a parallelogram are parallel. Properties of Parallelograms: 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.

8 If I give you 3 dots on a coordinate grid, how many different parallelograms could we make?

9 Homework: Practice 6-2A Change #12 to Prove: AB  CD and BC  AD