1. Find the value of x. ANSWER 60 2. Find the coordinates of the midpoint of AB with endpoints A(–2, 6) and B(10, 4). ANSWER (4, 5) 3. Find the distance between (–2, 6) and (10, 4). ANSWER 37 2

Use properties of parallelograms. Target Use properties of parallelograms. You will… Find angle and side measures in parallelograms.

Vocabulary parallelogram – a quadrilateral with both pairs of opposite sides parallel Properties of Parallelograms If a quadrilateral is a parallelogram, then opposite sides are congruent – Theorem 8.3 opposite angles are congruent – Theorem 8.4 consecutive angles are supplementary – Theorem 8.5 diagonals bisect each other – Theorem 8.6.

Use properties of parallelograms EXAMPLE 1 Use properties of parallelograms ALGEBRA Find the values of x and y. ABCD is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x and Theorem 8.4 to find the value of y. AB = CD Opposite sides of a are . x + 4 = 12 Substitute x + 4 for AB and 12 for CD. x = 8 Subtract 4 from each side. By Theorem 8.4, A C, or m A = m C. So, y ° = 65°. In ABCD, x = 8 and y = 65. ANSWER

GUIDED PRACTICE for Example 1 Find FG and m G. 1. 8, 60° ANSWER Find the values of x and y. 2. 25, 15 ANSWER

EXAMPLE 2 Use properties of parallelograms Desk Lamp As shown, part of the extending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. Find m BCD when m ADC = 110°. SOLUTION By Theorem 8.5, the consecutive angle pairs in ABCD are supplementary. So, m ADC + m BCD = 180°. Because m ADC = 110°, m BCD =180° –110° = 70°.

EXAMPLE 3 Standardized Test Practice SOLUTION By Theorem 8 .6, the diagonals of a parallelogram bisect each other. So, P is the midpoint of diagonals LN and OM . Use the Midpoint Formula. = Coordinates of midpoint P of OM 4 + 0 2 7 + 0 , ( ) = 7 ( ) The correct answer is A. ANSWER

GUIDED PRACTICE for Examples 2 and 3 Find the indicated measure in JKLM. NM 3. m JML 5. ANSWER 2 ANSWER 70° KM 4. m KML 6. ANSWER 4 ANSWER 40°