Question

given: ∠1 ≅ ∠2
prove: p || q

statements

1. ∠1 ≅ ∠2
2. ∠1 and ∠3 are vertical ∠s

∠3 ≅ ∠2

reasons

1. given
2. def. of vertical ∠s

converse of corresponding angles theorem

the statement is part of this proof, but at least one other statement must come before it
clear

Explanation:

Step1: Recall Vertical Angles Theorem

Vertical angles are congruent, so from step 2 ($\angle1$ and $\angle3$ are vertical $\angle$s), we get $\angle1\cong\angle3$.

Step2: Use Transitive Property

We know $\angle1\cong\angle2$ (given) and $\angle1\cong\angle3$ (vertical angles theorem). By transitive property of congruence, $\angle3\cong\angle2$.

Step3: Apply Converse of Corresponding Angles Theorem

Since $\angle3$ and $\angle2$ are corresponding angles and $\angle3\cong\angle2$, by converse of corresponding angles theorem, $p\parallel q$. But for the missing statement, after step 2, we use transitive property on $\angle1\cong\angle2$ and $\angle1\cong\angle3$ to get $\angle3\cong\angle2$.

Answer:

The missing statement is $\boldsymbol{\angle3\cong\angle2}$ (obtained by transitive property from $\angle1\cong\angle2$ and $\angle1\cong\angle3$).