Question

$\angle uvx \cong \angle qst$. complete the proof that $\overleftrightarrow{uw} \parallel \overleftrightarrow{rt}$.

	statement	reason
2	$\angle qst \cong \angle rsv$
3	$\angle uvx \cong \angle rsv$
4	$\overleftrightarrow{uw} \parallel \overleftrightarrow{rt}$

Explanation:

Step1: Identify Reason for Step 2

∠QST and ∠RSV are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for \( \angle QST \cong \angle RSV \) is "Vertical Angles Theorem".

Step2: Identify Reason for Step 3

We know \( \angle UVX \cong \angle QST \) (given) and \( \angle QST \cong \angle RSV \) (from step 2). By the Transitive Property of Congruence, if \( \angle A \cong \angle B \) and \( \angle B \cong \angle C \), then \( \angle A \cong \angle C \). So the reason for \( \angle UVX \cong \angle RSV \) is "Transitive Property of Congruence".

Step3: Identify Reason for Step 4

\( \angle UVX \) and \( \angle RSV \) are corresponding angles. If corresponding angles are congruent, then the lines cut by the transversal are parallel. So by the Corresponding Angles Converse, \( \overleftrightarrow{UW} \parallel \overleftrightarrow{RT} \). The reason is "Corresponding Angles Converse".

Answer:

1. Reason: Vertical Angles Theorem
2. Reason: Transitive Property of Congruence
3. Reason: Corresponding Angles Converse