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**6.3 Proving that a Quadrilateral is a Parallelogram**

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

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Theorem 6.9 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

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Theorem 6.10 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

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**Finding Values for Parallelograms**

For what value of y must PQRS be a parallelogram? 3x – 5 = 2x + 1 x – 5 = 1 x = 6 y = x + 2 y = 6 + 2 y = 8

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Theorem 6.11 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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Theorem 6.12 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

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**Deciding Whether a Quadrilateral is a Parallelogram**

Can you prove that the quadrilateral is a parallelogram based on the diagram? Explain. No. You need to show that both pairs of opposite sides are congruent, not consecutive sides. Yes, same-side interior angles are supplementary, making segment AB and segment DC parallel. Since segment AB is congruent to segment DC, ABCD is a parallelogram.

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**Concept Summary Proving that a Quadrilateral is a Parallelogram**

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More Practice!!!!! Homework – Textbook p. 372 #7 – 16 ALL.

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Sections 7.2-7.3. A parallelogram must have: Both pair of opposite sides congruent Both pair of opposite angles congruent Consecutive angles that.

Sections 7.2-7.3. A parallelogram must have: Both pair of opposite sides congruent Both pair of opposite angles congruent Consecutive angles that.

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