Question

complete the proof that ( mangle uty + mangle rwx = 180^{circ} ).

statement	reason
2 ( overleftrightarrow{eg}paralleloverleftrightarrow{vx} )	given
3 ( angle utycongangle gfy )	corresponding angles theorem
4 ( mangle gfy + mangle rwx = 180^{circ} )
5 ( mangle uty + mangle rwx = 180^{circ} )	substitution

Explanation:

Step1: Recall the Linear - Pair Postulate

If two angles form a linear pair, then the sum of their measures is \(180^{\circ}\).

Step2: Analyze \(\angle GFY\) and \(\angle RWX\)

Since \(\overleftrightarrow{EG}\parallel\overleftrightarrow{VX}\) (given), and \(\angle GFY\) and \(\angle RWX\) are same - side interior angles.
By the Same - Side Interior Angles Theorem (if two parallel lines are cut by a transversal, then the sum of the measures of a pair of same - side interior angles is \(180^{\circ}\)), we have \(m\angle GFY + m\angle RWX=180^{\circ}\)

Answer:

Same - Side Interior Angles Theorem