Exercise 26 Page 486

Let's draw three lines, $x,$ $y,$ and $z,$ and mark all of the angles they create.

Let's say we know two of them, $\angle 1$ and $\angle 3.$

Since the sides making up the angles are opposite rays, by the Vertical Angles Congruence Theorem we can identify both $\angle 4$ and $\angle 6.$

From the figure we see that $\angle 5$ is supplementary to $\angle 4$ and $\angle 6.$ Similarly, $\angle 2$ is supplementary to $\angle 1$ and $\angle 3.$ Since $\angle 5$ and $\angle 2$ are the only unknown angles in the two equations, we can find them by using the Substitution Property of Equality. Calculating these angles, we find that $\angle 2$ is congruent with $\angle 5.$ We also know this since $\angle 2$ and $\angle 5$ are vertical angles.