7.1 Properties of Parallelograms A parallelogram is a quadrilateral with both pairs of opposite sides parallel. In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.

Theorem 7-1-1 If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Example Find BD Find CD Find BE Find m<ABC Find m<ADC The picture below is a parallelogram. In ABCD, AB = 17.5, DE=18, and m<BCD = 110°. B C Find BD Find CD Find BE Find m<ABC Find m<ADC Find m<DAB E A D

Example 1 a) b) Y – 5 = 22 2x – 5 = 11 3x = 18 9x +1 = 28 + 5 = +5 3 3 X = 6 Y – 5 = 22 + 5 = +5 Y = 27 2x – 5 = 11 + 5 = +5 2x = 16 2 2 x = 8 9x +1 = 28 - 1 = - 1 9x = 27 9 9 x = 3

Theorem 7-1-2 If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Examples ABCD is a parallelogram. Find the m<B. C B (7y + 5)°

Example 2 a) b) (7x – 10)° (7x – 1)° 139° (6x – 1)° (7x – 1)° = 139° + 1 = +1 (7x)° = 140° (7x)° = 140° 7 7 x = 20 (7x – 10)° = (6x – 1)° -6x = -6x x – 10 = -1 +10 = +10 x = 9

Consecutive Angles Angles of a polygon that share a side are consecutive angles.

Theorem 7-1-3 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Using Consecutive Angles What is the measure of angle P in parallelogram PQRS? 26° 64° 116° 126°

Example 3 a) b) 120° (5x+20)° (8y +4)° 9x° 135° 9x° + 135° = 180° -135 = -135 9x° = 45° 9x = 45 9 9 x = 5 (5x+20)° + 120° = 180° 5x° + 140° = 180° - 140 = -140 5x° = 40° 5x = 40 5 5 x = 8 (8y+4)° + 120° = 180° 8y° + 124° = 180° - 140 = -140 8y° = 56° 8y = 56 8 8 x = 7

Theorem 7-1-4 If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Example Find b Find a b+8 = 5b 3a – 7 = 2a -b = -b + 7 = + 7 8 = 4b 4 4 b = 2 Find a 3a – 7 = 2a + 7 = + 7 3a = 2a + 7 -2a = -2a a = 7

Example 4 Find x 5x – 1 = 14 + 1 = +1 5x = 15 5x = 15 5 5 x = 3 Find y 18 5x - 1 14 9y b) 7y + 2 5x-8 10y - 7 4x + 4 Find x 5x – 1 = 14 + 1 = +1 5x = 15 5x = 15 5 5 x = 3 Find y 9y = 18 9y = 18 9 9 y = 2 Find x 5x – 8 = 4x +4 -4x = -4x x - 8 = 4 +8 = +8 x = 12 Find y 10y – 7 = 7y + 2 -7y = -7y 3y – 7 = 2 +7 = +7 3y = 9 3y = 9 3 3 y = 3