Question

$\angle vyx \cong \angle vwx$ and $\angle wvy \cong \angle wxy$. complete the proof that $\overline{wx} \parallel \overline{vy}$.

statement	reason
2 $\angle wvy \cong \angle wxy$	given
3 $m\angle wvy + m\angle vwx + m\angle wxy + m\angle vyx = 360^\circ$	quadrilateral angle sum theorem
4 $m\angle wvy + m\angle vwx + m\angle wvy + m\angle vwx = 360^\circ$	substitution
5 $m\angle wvy + m\angle vwx = 180^\circ$	properties of addition, subtraction, multiplication, and division
6

Explanation:

Step1: Identify the consecutive interior angles

We know from step 5 that \( m\angle WVY + m\angle VWX = 180^\circ \). Notice that \( \angle WVY \) and \( \angle VWX \) are consecutive interior angles formed by the transversal \( VW \) intersecting lines \( WX \) and \( VY \).

Step2: Apply the Consecutive Interior Angles Theorem

The Consecutive Interior Angles Theorem states that if two consecutive interior angles are supplementary (their measures sum to \( 180^\circ \)), then the two lines cut by the transversal are parallel. Since \( m\angle WVY + m\angle VWX = 180^\circ \), by the Consecutive Interior Angles Theorem, we can conclude that \( \overline{WX} \parallel \overline{VY} \).

Answer:

Statement 6: \( \overline{WX} \parallel \overline{VY} \)
Reason 6: Consecutive Interior Angles Theorem (If two consecutive interior angles are supplementary, then the lines are parallel)