Question

complete the proof that $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$.
statement reason
1 $angle svwcongangle rsv$ given
2 $angle qstcongangle rsv$ vertical angle theorem
3 $angle svwcongangle qst$
4 $overleftrightarrow{uw}paralleloverleftrightarrow{rt}$

Explanation:

Step1: Recall vertical - angle property

$\angle QST$ and $\angle RSV$ are vertical angles. By the Vertical Angle Theorem, vertical angles are congruent. So, $\angle QST\cong\angle RSV$.

Step2: Use transitive property of congruence

We know that $\angle SVW\cong\angle RSV$ (given) and $\angle QST\cong\angle RSV$. By the transitive property of congruence, if $\angle A\cong\angle B$ and $\angle C\cong\angle B$, then $\angle A\cong\angle C$. So, $\angle SVW\cong\angle QST$.

Step3: Apply corresponding - angles postulate

$\angle SVW$ and $\angle QST$ are corresponding angles. If corresponding angles are congruent, then the two lines are parallel. Since $\angle SVW\cong\angle QST$, we can conclude that $\overleftrightarrow{UW}\parallel\overleftrightarrow{RT}$.

Answer:

1. Reason: Vertical Angle Theorem; 3. Reason: Transitive property of congruence; 4. Reason: Corresponding - angles postulate.