6.3 Proving Quadrilaterals are Parallelograms Goals p Prove that a quadrilateral is a parallelogram. p Use coordinate geometry with parallelograms. THEOREM 6.6 If both pairs of opposite sides of a A B quadrilateral are congruent, then the quadrilateral is a parallelogram. DC ABCD is a parallelogram. THEOREM 6.7 If both pairs of opposite angles of a A B quadrilateral are congruent, then the quadrilateral is a parallelogram. DC ABCD is a parallelogram. THEOREM 6.8 If an angle of a quadrilateral is A B (180 Ϫ x)Њ xЊ supplementary to both of its consecutive angles, then the xЊ quadrilateral is a parallelogram. DC ABCD is a parallelogram. THEOREM 6.9 If the diagonals of a quadrilateral A B bisect each other, then the M quadrilateral is a parallelogram. DC ABCD is a parallelogram. Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 6 • Geometry Notetaking Guide 124 Example 1 Proof of Theorem 6.8 Prove Theorem 6.8. K L Given: aJ is supplementary to aK and aM. Prove: JKLM is a parallelogram. JM Statements Reasons 1. aJ is supplementary to aK. 1. Given 2. JM& ʈ KL& 2. Consecutive Interior A Converse 3. aJ is supplementary to aM. 3. Given 4. JK& ʈ ML&& 4. Consecutive Interior A Converse 5. JKLM is a parallelogram. 5. Definition of a parallelogram Checkpoint Complete the following exercise. 1. Stained Glass A pane in a stained glass 21 cm window has the shape shown at the right. How do you know that the pane is a 45 cm parallelogram? 45 cm The pane is a quadrilateral and both pairs of opposite sides are congruent. So, by Theorem 6.6, the quadrilateral is a 21 cm parallelogram. THEOREM 6.10 If one pair of opposite sides of a A B quadrilateral are congruent and parallel ,then the quadrilateral is a parallelogram. DC ABCD is a parallelogram. Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 6 • Geometry Notetaking Guide 125 Example 2 Using Properties of Parallelograms Show that A(1, 3), B(3, 5), C(9, 1), y B(3, 5) and D(7, Ϫ1) are the vertices of 5 a parallelogram. 3 A(1, 3) Solution C(9, 1) 1 To find the Method 1 Show that opposite sides 13579x slopes of the have the same slope. opposite sides, use D(7, Ϫ1) rise slope ϭ ᎏᎏ Slope of AB‡: Slope of BC‡‡: run Ϫ y2 y1 5 Ϫ 3 5 Ϫ 1 ϭ ᎏᎏ. Ϫᎏ2ᎏ x Ϫ x 1 2 1 3 Ϫ 1 3 Ϫ 9 3 Slope of CD‡: Slope of AD‡: Ϫ Ϫ Ϫ ΃ 1 ( 1) 3 ΂ Ϫ1 2 1 ϭ Ϫᎏᎏ 9 Ϫ 7 1 Ϫ 7 3 ᭤ AB‡ and CD‡‡ have the same slope, so they are parallel. Similarly, BC‡‡ ʈ AD‡‡ . Because opposite sides are parallel , ABCD is a parallelogram. Method 2 Show that opposite sides have the same length. AB ϭ ͙(ෆ 3 Ϫෆ1)2 ϩෆ( 5 Ϫෆ3 )2 ϭ ͙8ෆ 2͙2ෆ To find the lengths of the sides, BC ϭ ͙(9ෆ Ϫ 3ෆ )2 ϩෆ( 1 Ϫෆ5 )2 ϭ ͙52ෆ 2͙13ෆ use the Distance Formula. CD ϭ ͙(ෆ 9 Ϫෆ7)2 ϩෆ΄ 1 Ϫෆ( Ϫ1 )ෆ΅2 ϭ ͙8ෆ 2͙2ෆ AD ϭ ͙(ෆ 7 Ϫෆ1 )2 ϩෆ(Ϫ1ෆϪ 3 )ෆ2 ϭ ͙52ෆ 2͙13ෆ ᭤ AB‡c CD‡‡ and BC‡‡c AD‡‡ . 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. Slope of AB‡ ϭ Slope of CD‡ 1 AB ϭ CD 2͙2ෆ ᭤ AB‡ and CD‡ are congruent and parallel . So, ABCD is a parallelogram. Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 6 • Geometry Notetaking Guide 126.