1. Objectives
State the conditions under which you can prove a quadrilateral is a parallelogram

2. Converse of Theorem 6-1
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

3. Converse of Theorem 6-2
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

4. Converse of Theorem 6-3
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

5. Proof of the Converse of Thm 6-3

6. Theorem 6-8
If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

7. Example 1
For value of x will quadrilateral MNPL be a parallelogram?
If the diagonals bisect each other, then the quadrilateral is a parallelogram.
2y – 7 = y + 2
y – 7 =
y = 9
3x = y
3x = 9
x = 3

8. Example 2a
Angles A and C are congruent. ∠ADC and ∠CBA are congruent by the Angle Addition Postulate. Since both pairs of opposite angles are congruent, ABCD is a parallelogram.

9. Example 2b
This cannot be proven because there is not enough information given. It is not stated that the single-marked sides are congruent to the double-marked sides. If opposite sides are congruent, then the quadrilateral is a parallelogram.

10. Quick Check 2a
Since one pair of opposite sides are both parallel and congruent, we can use Theorem 6-8 to prove PQRS is a parallelogram.

11. Quick Check 2b
Not enough information is given. It is not stated that the single-marked segments are congruent to the double-marked segments. If diagonals bisect each other, then the quadrilateral is a parallelogram.