This document discusses several theorems for proving that a quadrilateral is a parallelogram: 1) If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. 2) If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. 3) If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram. The document also provides proofs of these theorems and discusses using coordinate geometry to prove sides are congruent or parallel.

1. 6.3 Proving Quadrilaterals are Parallelograms Objective: Prove that a quadrilateral is a parallelogram Use coordinate geometry with parallelograms.

2. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem

3. Theorem If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

4. Theorem If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. x˚ 180 - x˚ x˚

5. Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

6. Proof Prove Theorem 6.6 - If both pairs of opposite sides are  , then the quadrilateral is a parallelogram. Given: AB  CD and AD  CB Prove: ABCD is a parallelogram A B C D

7. Proof AB  CD and AD  CB AC  AC ∆ ABC  ∆CDA <BAC  <DCA, <DAC  <BCA AB || CD, AD || CB ABCD is a parallelogram. Given Reflexive SSS CPCTC Alternate Interior Angles Converse Definition of a parallelogram A B C D

8. Theorem If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram

9. Summary - How can we prove quadrilaterals are parallelograms? Show that both pairs of opposite sides are || Show that both pairs of opposite sides are  Show that both pairs of opposite angles are  Show that one angle is supplementary to both consecutive angles Show that the diagonals bisect each other Show that one pair of opposite sides are  and ||

10. Using Coordinate Geometry When a figure is in the coordinate plane, you can use the Distance Formula to prove that the sides are congruent and you can use the slope formula to prove that sides are parallel. Go through Example 4 on p. 341 Do p. 342 1-8

11. Homework: p. 342 10-24 evens