Question

given: x || y
w is a transversal of x and y.
prove: ∠4 ≅ ∠5
complete the two - column proof.
a
b

Explanation:

Step1: Vertical Angles Theorem

Vertical angles are congruent. $\angle4$ and $\angle1$ are vertical angles, so $\angle4\cong\angle1$.

Step2: Corresponding Angles Theorem

If two parallel lines are cut by a transversal, then corresponding angles are congruent. Since $x\parallel y$ and $w$ is a transversal, $\angle1$ and $\angle5$ are corresponding angles, so $\angle1\cong\angle5$.

Step3: Transitive Property of Congruence

If $a\cong b$ and $b\cong c$, then $a\cong c$. Here $a = \angle4$, $b=\angle1$, $c = \angle5$. Since $\angle4\cong\angle1$ and $\angle1\cong\angle5$, by transitive property $\angle4\cong\angle5$.

Answer:

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