Exercise 42 Page 292

In this solution, we used the fact that vertical angles are always congruent. This is true due the Vertical Angle Theorem. We will learn a bit more about this theorem by considering the next diagram.

Analyzing the diagram, it can be observed that $\angle 1$ and $\angle 2$ form a straight angle, so these are supplementary angles. Similarly, $\angle 2$ and $\angle 3$ are also supplementary angles.

Therefore, by the Angle Addition Postulate, the sum of $m\angle 1$ and $m\angle 2$ is $180^{\circ },$ and the sum of $m\angle 2$ and $m\angle 3$ is also $180^{\circ }.$ These facts can be used to express $m\angle 2$ in terms of $m\angle 1$ and also in terms of $m\angle 3.$

Angle Addition Postulate	Isolate $m\angle 2$
$m\angle 1+m\angle 2$ $=$ $180^{\circ }$	$m\angle 2$ $=$ $180^{\circ }-m\angle 1$
$m\angle 2+m\angle 3$ $=$ $180^{\circ }$	$m\angle 2$ $=$ $180^{\circ }-m\angle 3$

By the Transitive Property of Equality, the expressions representing $m\angle 2$ can be set equal to each other. Then the obtained equation can be simplified. By the definition of congruent angles, this means that the vertical angles $\angle 1$ and $\angle 3$ are congruent angles. Using the same argumentation, $\angle 2$ and $\angle 4$ can also be proven to be congruent.