6.1.2 Properties of Parallelograms Key Objectives • Prove and apply properties of parallelograms. • Use properties of parallelograms to solve problems. Key Terms
• A quadrilateral with two pairs of parallel sides is a parallelogram . Theorems, Postulates, Corollaries, and Properties
• Theorem If a quadrilateral is a parallelogram, then its opposite sides are congruent. • Theorem If a quadrilateral is a parallelogram, then its opposite angles are congruent. • Theorem If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. • Theorem If a quadrilateral is a parallelogram, then its diagonals bisect each other. Example 1 Problem-Solving Application
The sides opposite each other in a parallelogram are congruent.
Consecutive angles, those on either end of a single side, in a parallelogram are supplementary, meaning their measures sum to 180°.
The length of a side and measure of an angle in a parallelogram are determined in this example. AD , m ∠ CDA , and AE are given. To find BC , recognize that BC ≅ AD by the Properties of Parallelograms (opposite sides are congruent). Then, by the definition of congruent line segments, BC = AD . Substitute the known length for AD to find BC = 60 in. To find m ∠ DAB , recognize that ∠ CDA is consecutive with ∠ DAB . Therefore, the sum of their measures equals 180°, according to the Properties of Parallelograms. Substitute the known value of 81° for m ∠ CDA . The measure of ∠ DAB is found to be 99°.
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