Question

in the figure below, the measure of ∠4 = 90°, the measure of ∠5 = 53°, and the measure of ∠6 = 37°. what are the measures of ∠1, ∠2, and ∠3? m∠1 = 90° m∠2 = □°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle4$ are vertical angles. Since $\angle4 = 90^{\circ}$, then $m\angle1=90^{\circ}$.

Step2: Use angle - sum property of a straight - line

The sum of angles on a straight - line is $180^{\circ}$. $\angle2+\angle4+\angle5 = 180^{\circ}$. We know $\angle4 = 90^{\circ}$ and $\angle5 = 53^{\circ}$.
So, $m\angle2=180^{\circ}-m\angle4 - m\angle5$.
Substitute the values: $m\angle2=180 - 90-53$.
$m\angle2 = 37^{\circ}$.

Step3: Use vertical - angle property for $\angle3$

$\angle3$ and $\angle5$ are vertical angles. So, $m\angle3=m\angle5 = 53^{\circ}$.

Answer:

$m\angle2 = 37^{\circ}$, $m\angle3 = 53^{\circ}$