Question

in the figure below, the measure of ∠4 = 90°, the measure of ∠5 = 55°, and the measure of ∠6 = 35°. what are the measures of ∠1, ∠2, and ∠3?

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle1$ and $\angle5$ are vertical angles, $\angle2$ and $\angle6$, $\angle3$ and $\angle7$, $\angle4$ and $\angle8$ (not in our problem but for general understanding). Since $\angle1$ and $\angle5$ are vertical angles, $\angle1=\angle5 = 55^{\circ}$.

Step2: Use linear - pair property

$\angle1+\angle2+\angle4 = 180^{\circ}$ (a straight - line angle measures $180^{\circ}$). Given $\angle4 = 90^{\circ}$ and $\angle1 = 55^{\circ}$. Then $\angle2=180^{\circ}-\angle1 - \angle4$.
$\angle2=180^{\circ}-55^{\circ}-90^{\circ}=35^{\circ}$.

Step3: Use vertical - angle property for $\angle7$

$\angle3$ and $\angle7$ are vertical angles. Also, $\angle3 = \angle1$ (vertical angles). So $\angle7=\angle1 = 55^{\circ}$.

Answer:

$\angle2 = 35^{\circ}$, $\angle7 = 55^{\circ}$