Question

proving alternate interior angles congruent
given ( x parallel y )
( w ) is a transversal of ( x ) and ( y ).
prove: ( angle 4 cong angle 5 )
complete the two - column proof.

Explanation:

Step1: Vertical angles are congruent

Vertical angles are formed by two intersecting lines. $\angle4$ and $\angle1$ are vertical angles. By the vertical angles theorem, $\angle4\cong\angle1$.

Step2: Corresponding angles are congruent

Since $x\parallel y$ and $w$ is a transversal, $\angle1$ and $\angle5$ are corresponding angles. By the corresponding angles postulate, if two parallel lines are cut by a transversal, then corresponding angles are congruent. So, $\angle1\cong\angle5$.

Step3: Transitive property of congruence

If $\angle4\cong\angle1$ and $\angle1\cong\angle5$, then by the transitive property of congruence (if $a = b$ and $b = c$, then $a = c$ for congruent angles), $\angle4\cong\angle5$.

Answer:

A: Vertical Angles Theorem; B: Corresponding Angles Postulate; C: Transitive Property of Congruence.