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LESSON 6.3 NAME _________________________________________________________ DATE ____________ Practice with Examples For use with pages 338–346 GOAL Prove that a quadrilateral is a parallelogram and use coordinate geometry with parallelograms Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Chapter 6 Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Ways to Prove a Shape is a Parallelogram • Show that both pairs of opposite sides are parallel. • Show that both pairs of opposite sides are congruent. • Show that both pairs of opposite angles are congruent. • Show that one angle is supplementary to both consecutive angles. • Show that the diagonals bisect each other. • Show that one pair of opposite sides are congruent and parallel. 106 Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved. LESSON 6.3 CONTINUED EXAMPLE 1 NAME _________________________________________________________ DATE ____________ Practice with Examples For use with pages 338–346 Using Properties of Parallelograms Show that A2, 0, B3, 4, C2, 6, and D3, 2 are the vertices of a parallelogram. y C SOLUTION B There are many ways to solve this problem. Slope of AB 40 4 32 Slope of CD 26 4 4 3 2 1 Slope of BC 64 2 2 2 3 5 5 Slope of DA 02 2 2 2 3 5 5 2 Chapter 6 Method 1 Show that opposite sides have the same slope, so they are parallel. D A 1 x AB and CD have the same slope, so they are parallel. Similarly, BC DA. Because opposite sides are parallel, ABCD is a parallelogram. Method 2 Show that the opposite sides have the same length. AB 3 22 4 02 17 CD 3 22 2 62 17 BC 2 3)2 6 42 29 DA 2 32 0 22 29 AB CD and BC DA. Because both pairs of opposite sides are congruent, ABCD is a parallelogram. Method 3 Show that one pair of opposite sides is congruent and parallel. Find the slopes and lengths of AB and CD as shown in Methods 1 and 2. Slope of AB Slope of CD 4 AB CD 17 AB and CD are congruent and parallel, so ABCD is a parallelogram. Copyright © McDougal Littell Inc. All rights reserved. Geometry Practice Workbook with Examples 107 LESSON 6.3 CONTINUED NAME _________________________________________________________ DATE ____________ Practice with Examples For use with pages 338–346 Exercises for Example 1 Refer to the methods demonstrated in Example 1 to show that the quadrilateral with the given vertices is a parallelogram. 1. Show that the quadrilateral with vertices A3, 0, B2, 4, Chapter 6 C7, 6, and D8, 2 is a parallelogram using Method 1 from Example 1. 2. Show that the quadrilateral with vertices A4, 1, B1, 2, C4, 4, and D1, 5 is a parallelogram using Method 2 from Example 1. 3. Show that the quadrilateral with vertices A0, 6, B4, 5, C6, 3, and D2, 2 is a parallelogram using Method 3 from Example 1. 4. Show that the quadrilateral with vertices A1, 2, B5, 3, C6, 6, and D0, 7 is a parallelogram using any of the three methods demonstrated in Example 1. 108 Geometry Practice Workbook with Examples Copyright © McDougal Littell Inc. All rights reserved.