6.1.2 Properties of Parallelograms (continued) Example 2 Using Properties of Parallelograms to Find Measures
The measures of a side and an angle in a parallelogram are determined in this example. AB and DC , m ∠ B , and m ∠ A are given as algebraic expressions. AB and DC are equal according to the Properties of Parallelograms. Set the expressions for the lengths of the two sides equal to each other and solve for x .The solution yields x = 7, and substituting this value for x in the expression for AB yields a length of 19 units. ∠ B and ∠ A are consecutive angles in a parallelogram, so according to the Properties of Parallelograms their measures sum to 180°. Substitute the expressions for m ∠ B and m ∠ A into the equation for their sum and solve for the unknown, y . The solution yields y = 11, and substituting this value for y in the expression for the measure of ∠ B yields 70°.
Example 3 Parallelograms in the Coordinate Plane
The coordinates are determined here for the unknown vertex of a parallelogram on the coordinate plane. The coordinates of three of the vertices for the parallelogram are given. The fourth vertex, D , will form the endpoint of a line segment, CD , that is parallel to line segment AB . The slope of the two sides will be equal, since this is a parallelogram. The length of the two sides will be equal according to the Properties of Parallelograms. Use the slope of AB , a rise of 3 units and a run of 1 unit, to find point D relative to point C . For the x coordinate, 2 + 1 = 3, and for the y coordinate, − 4 + 3 = − 1. The coordinates for D are (3, − 1).
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