Geometry: Common Core (15th Edition)

Theorem 2.2, the Congrent Supplement Theorem, tells is that if two angles are supplements of congruent angles, then those two angles are congruent. $\angle1$ and $\angle3$ are supplementary because they form a linear pair. $\angle2$ and $\angle4$ are supplementary because they form a linear pair. Since $\angle1\cong\angle2$, Theorem 2.2 tells us that $\angle3\cong\angle4$.