Question

uw || rt. complete the proof that ∠uvx ≅ ∠qst.
statement:

1. uw || rt
2. ∠uvx ≅ ∠rsv
3. ∠rsv ≅ ∠qst
4. ∠uvx ≅ ∠qst

reason:
choices: angles forming a linear pair sum to 180°, converse of corresponding angles theorem, corresponding angles theorem, definition of congruence, definition of supplementary angles

Explanation:

Step1: Identify Given Information

Given \(\overleftrightarrow{UW} \parallel \overleftrightarrow{RT}\) (Statement 1). We need to prove \(\angle UNX \cong \angle QST\) (Statement 4) by filling in reasons for Statements 2 and 3.

Step2: Reason for Statement 2 (\(\angle UNX \cong \angle RSV\))

Since \(\overleftrightarrow{UW} \parallel \overleftrightarrow{RT}\) and they are cut by a transversal (line \(XVQS\)), \(\angle UNX\) and \(\angle RSV\) are corresponding angles. By the Corresponding Angles Theorem, if two parallel lines are cut by a transversal, corresponding angles are congruent. So the reason for Statement 2 is "Corresponding Angles Theorem".

Step3: Reason for Statement 3 (\(\angle RSV \cong \angle QST\))

\(\angle RSV\) and \(\angle QST\) are vertical angles (opposite angles formed by intersecting lines \(XVQS\) and \(\overleftrightarrow{RT}\)). By the Definition of Congruence for vertical angles (vertical angles are congruent), the reason for Statement 3 is "Definition of Congruence" (or "Vertical Angles Congruence Theorem", which is a specific case of congruence definition).

Step4: Reason for Statement 4 (\(\angle UNX \cong \angle QST\))

By the Transitive Property of Congruence (if \(\angle A \cong \angle B\) and \(\angle B \cong \angle C\), then \(\angle A \cong \angle C\)), since \(\angle UNX \cong \angle RSV\) (Statement 2) and \(\angle RSV \cong \angle QST\) (Statement 3), we conclude \(\angle UNX \cong \angle QST\). But in the given reasons, we can also trace the steps: Statement 2 uses Corresponding Angles Theorem, Statement 3 uses Definition of Congruence (vertical angles), and then transitivity. However, the key steps for the proof structure are filling the reasons for each statement as above.

Answer:

To complete the proof:

• Reason for Statement 1 (\(\overleftrightarrow{UW} \parallel \overleftrightarrow{RT}\)): Given (not in the reason choices, as it's the premise).
• Reason for Statement 2 (\(\angle UNX \cong \angle RSV\)): \(\boldsymbol{\text{Corresponding Angles Theorem}}\)
• Reason for Statement 3 (\(\angle RSV \cong \angle QST\)): \(\boldsymbol{\text{Definition of Congruence}}\) (or "Vertical Angles Congruence Theorem")
• Reason for Statement 4 (\(\angle UNX \cong \angle QST\)): Transitive Property of Congruence (implied by Statements 2 and 3).

(If choosing from the given reason list for each step:

• Statement 2: "Corresponding Angles Theorem"
• Statement 3: "Definition of Congruence"
• Statement 4: Follows from transitivity, but the proof structure connects the angles via corresponding angles and vertical angles congruence.)