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# ​ Quadrilateral ABCD ​ is inscribed in a circle. What is the measure of angle A?

May 12, 2017

$m \angle A = {77}^{\circ}$

#### Explanation:

Consider the property of cyclic quadrilaterals for which opposite angles are supplementary, then:
$m \angle A + m \angle C = {180}^{\circ}$

Since we are given that $m \angle A = {\left(2 x + 9\right)}^{\circ}$ and $m \angle C = {\left(3 x + 1\right)}^{\circ}$, we can substitute to find the value of $x$:
$2 x + 9 + 3 x + 1 = 180$
$5 x + 10 = 180$
$5 x = 170$
$x = 34$

Now, we can substitute in the above value of $x$ to find $m \angle A$:
$m \angle A = {\left(2 x + 9\right)}^{\circ} = {\left(2 \cdot 34 + 9\right)}^{\circ} = {77}^{\circ}$

Therefore, the $m \angle A = {77}^{\circ}$