6.1.3 Conditions for Parallelograms (continued)
In this example, the quadrilateral ABCD is shown to be a parallelogram when the unknowns x and y have certain values. The lengths of the sides of the quadrilateral are given as algebraic expressions in x and y . Substitute the given values for x and y into the expressions for the lengths of the sides, solve for the lengths, and compare opposite sides. Since BC = DA = 15, and AB = CD = 11, then by the quadrilateral with congruent opposite sides condition for parallelograms, the figure is a parallelogram. The quadrilateral EFGH is shown here to be a parallelogram when the unknowns z and w have certain values. The measures of three of the angles of the quadrilateral are given as algebraic expressions in z and w . Substitute the given values for z and w into the expressions for the angle measures, and solve for the measures. This proof is more direct than the one shown. ∠ F and ∠ G are consecutive angles. Since m ∠ F + m ∠ G = 98° + 82° = 180°, ∠ F and ∠ G are supplementary. Therefore, EFGH is a parallelogram by the Properties of Parallelograms (Quad. with ∠ s supp. to cons. ∠ s → ▱ ). The same proof can be obtained using ∠ F and ∠ E .
Example 2 Applying Conditions for Parallelograms
This is a condition of a quadrilateral being a parallelogram. If a quadrilateral has one pair of opposite congruent and parallel sides, then it is a parallelogram. Note the symbols for congruent sides and those for parallel sides in the figure.
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