# Chapter 4 - Section 4.1 - Properties of a Parallelogram - Exercises - Page 188: 13

$m\angle A = 80^{\circ}$ $m\angle B = 100^{\circ}$ $m\angle C = 80^{\circ}$ $m \angle D = 100^{\circ}$

#### Work Step by Step

The opposite angles of a parallelogram have equal measures. We can find the value of $x$: $m \angle A = m \angle C$ $\frac{2x}{3} = \frac{x}{2}+20$ $\frac{2x}{3} - \frac{x}{2} = 20$ $\frac{4x}{6} - \frac{3x}{6} = 20$ $\frac{x}{6} = 20$ $x = 120$ We can find the measure of $\angle A$: $m \angle A = \frac{2x}{3} = \frac{(2)(120)}{3} = 80^{\circ}$ We can find the measure of $\angle C$: $m \angle C = \frac{x}{2}+20 = \frac{(120)}{2}+20 = 80^{\circ}$ Let $a = m\angle B = m \angle D$ The sum of the four angles in a parallelogram is $360^{\circ}$ We can find the value of $a$: $80^{\circ}+a+80^{\circ}+a = 360^{\circ}$ $2a+160^{\circ} = 360^{\circ}$ $2a = 360^{\circ}-160^{\circ}$ $2a = 200^{\circ}$ $a = 100^{\circ}$ Then: $m\angle B = m \angle D = 100^{\circ}$

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