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**If x = 18 and y = 89, then ABCD is a parallelogram.**

Proving That a Quadrilateral is a Parallelogram LESSON 6-3 Additional Examples Find values of x and y for which ABCD must be a parallelogram. If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other. 10x – 24 = 8x Diagonals of parallelograms 2y – 80 = y + 9 bisect each other. 2x – 24 = y – 80 = 9 Collect the variable terms on one side. x = 18 2x = 36 y = 89 Solve. If x = 18 and y = 89, then ABCD is a parallelogram. Quick Check

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**Determine whether the quadrilateral is a parallelogram. Explain.**

Proving That a Quadrilateral is a Parallelogram LESSON 6-3 Additional Examples Determine whether the quadrilateral is a parallelogram. Explain. a. All you know about the quadrilateral is that only one pair of opposite sides is congruent. a. Therefore, you cannot conclude that the quadrilateral is a parallelogram. b. b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360. If x represents the measure of the unmarked angle, x = 360, so x = 105. Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-6. Quick Check

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**Proving That a Quadrilateral is a Parallelogram**

LESSON 6-3 Additional Examples The captain of a fishing boat plots a course toward a school of bluefish. One side of a parallel rule connects the boat with the school of bluefish. The other side makes a 36° angle north of due east on the chart’s compass. Explain how the captain knows in which direction to sail to reach the bluefish. Because both sections of the rulers and the crossbars are congruent, the rulers and crossbars form a parallelogram. Therefore, the angle shown on the chart’s compass is congruent to the angle the boat should travel, which is 36° north of due east. Quick Check

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