Question

complete the proof that (overleftrightarrow{wy}paralleloverleftrightarrow{tv}).
statement reason
1 (overleftrightarrow{tv}paralleloverleftrightarrow{gi}) given
2 (overleftrightarrow{wy}paralleloverleftrightarrow{gi}) given
3 (angle{sut}congangle{ghs}) corresponding angles theorem
4 (angle{ghs}congangle{sxw})
5 (angle{sut}congangle{sxw})
6 (overleftrightarrow{wy}paralleloverleftrightarrow{tv}) alternate exterior angles theorem; alternate interior angles theorem; converse of alternate exterior angles theorem; converse of alternate interior angles theorem; converse of corresponding angles theorem

Explanation:

Step1: Recall angle - relationship concepts

We know that when two lines are parallel to the same line, we can use angle - congruence relationships to prove other lines are parallel.

Step2: Analyze the given parallel lines

We are given $\overleftrightarrow{TV}\parallel\overleftrightarrow{GI}$ and $\overleftrightarrow{WY}\parallel\overleftrightarrow{GI}$. By the Corresponding Angles Theorem, when $\overleftrightarrow{TV}\parallel\overleftrightarrow{GI}$, $\angle{SUT}\cong\angle{GHS}$.

Step3: Identify the correct angle - congruence reason for $\angle{GHS}\cong\angle{SXW}$

$\angle{GHS}$ and $\angle{SXW}$ are vertical angles. Vertical angles are always congruent.

Step4: Use the transitive property of congruence

Since $\angle{SUT}\cong\angle{GHS}$ and $\angle{GHS}\cong\angle{SXW}$, by the transitive property of congruence, $\angle{SUT}\cong\angle{SXW}$.

Step5: Apply the parallel - line theorem

$\angle{SUT}$ and $\angle{SXW}$ are corresponding angles. If corresponding angles are congruent, then the lines are parallel. The reason for $\overleftrightarrow{WY}\parallel\overleftrightarrow{TV}$ is the Converse of Corresponding Angles Theorem.

Answer:

Converse of Corresponding Angles Theorem