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Objectives

• State the conditions under which you can prove a quadrilateral is a parallelogram

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Converse of Theorem 6-1

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

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Converse of Theorem 6-2

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

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Converse of Theorem 6-3

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

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Proof of the Converse of Thm 6-3

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Theorem 6-8

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

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Example 1

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

2y – 7 = y + 2 y – 7 = 2 y = 93x = y3x = 9 x = 3

For value of x will quadrilateral MNPL be a parallelogram?

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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.

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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.

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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.

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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.