Question

find the measures of the numbered angles in the figure, shown to the right. name the theorem that justifies each answer

Explanation:

Step1: Identify vertical - angles

Vertical angles are equal. The angle vertical to the $120^{\circ}$ angle has measure $120^{\circ}$. Let's call the angle adjacent to the $120^{\circ}$ angle $\angle A$. Since a straight - line has an angle measure of $180^{\circ}$, if one angle is $120^{\circ}$, then $\angle A=180 - 120=60^{\circ}$. The angle vertical to $\angle A$ also has measure $60^{\circ}$.
For the $137^{\circ}$ angle, the vertical angle to it has measure $137^{\circ}$. Let's call the angle adjacent to the $137^{\circ}$ angle $\angle B$. Then $\angle B = 180-137 = 43^{\circ}$, and the angle vertical to $\angle B$ has measure $43^{\circ}$.
Assume the top - left angle is $120^{\circ}$.
The angle adjacent to it (let's say $\angle1$) forms a linear pair.
$180 - 120=60^{\circ}$, so $\angle1 = 60^{\circ}$ (Linear - Pair Theorem: Two adjacent angles that form a straight line are supplementary, i.e., their sum is $180^{\circ}$).
The angle vertical to the $137^{\circ}$ angle (let's say $\angle2$) is $137^{\circ}$ (Vertical - Angles Theorem: Vertical angles are congruent).
The angle adjacent to the $137^{\circ}$ angle (let's say $\angle3$) is $180 - 137=43^{\circ}$ (Linear - Pair Theorem).
The angle vertical to $\angle3$ (let's say $\angle4$) is $43^{\circ}$ (Vertical - Angles Theorem).

Answer:

If we assume the angles are numbered in a logical order starting from the top - left:
The angle adjacent to the $120^{\circ}$ angle has measure $60^{\circ}$ (by the Linear - Pair Theorem).
The angle vertical to the $137^{\circ}$ angle has measure $137^{\circ}$ (by the Vertical - Angles Theorem).
The angle adjacent to the $137^{\circ}$ angle has measure $43^{\circ}$ (by the Linear - Pair Theorem).
The angle vertical to the $43^{\circ}$ angle has measure $43^{\circ}$ (by the Vertical - Angles Theorem).