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**Recall Theorem 6-3 The diagonals of a parallelogram bisect each other.**

Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals to solve a real-world problem Objectives: Determine whether a quadrilateral is a parallelogram Recall Theorem 6-3 The diagonals of a parallelogram bisect each other.

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**If a diagram shows two diagonals bisect then it is a parallelogram.**

Theorem 6-5 Converse of Theorem 6-3 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If a diagram shows two diagonals bisect then it is a parallelogram.

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Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

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**Use Theorems to Solve Problems**

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.

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Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

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Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.

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**Use Theorems to Solve Problems**

Determine whether the quadrilateral is a parallelogram. Explain. a. a. All you know about the quadrilateral is that only one pair of opposite sides is congruent. 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. Theorem 6-8 states If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-8.

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**Parallelograms and Real-world**

Dallas Center Waterfront Park in Vancouver B.C.

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