Geometry: Common Core (15th Edition)

$\angle1$ and $\angle3$ are vertical angles because it is given. $\angle1$ and $\angle2$ are supplementary by the definition of a linear pair. $\angle2$ and $\angle3$ are also supplementary by the definition of a linear pair. By the definition of supplementary, $m\angle1+m\angle2=180$ and $m\angle2+m\angle3=180$. By the Transitive Property of Equality, $m\angle1+m\angle2=m\angle2+m\angle3.$ By the Subtraction Property of Equality $m\angle1=m\angle3.$ $\angle1\cong\angle2$ because angles with the same measure are congruent.