Question

complete the proof that (mangle rw v + mangle sty=180^{circ}).
statement\treason
1 (overleftrightarrow{su}paralleloverleftrightarrow{fh})\tgiven
2 (overleftrightarrow{vx}paralleloverleftrightarrow{fh})\tgiven
3 (angle rw vcongangle fgr)\t
4 (mangle fgr + mangle sty = 180^{circ})\t
5 (mangle rw v + mangle sty = 180^{circ})\t

Explanation:

Step1: Identify corresponding - angles property

Since $\overleftrightarrow{SU}\parallel\overleftrightarrow{FH}$ and $\overleftrightarrow{VX}\parallel\overleftrightarrow{FH}$, then $\overleftrightarrow{SU}\parallel\overleftrightarrow{VX}$. $\angle RWV$ and $\angle FGR$ are corresponding angles (by the parallel - line properties), so $\angle RWV\cong\angle FGR$ (corresponding angles of parallel lines are congruent).

Step2: Use linear - pair property

$\angle FGR$ and $\angle STY$ form a linear pair. By the linear - pair postulate, the sum of the measures of angles in a linear pair is $180^{\circ}$, so $m\angle FGR + m\angle STY=180^{\circ}$.

Step3: Apply substitution property

Since $\angle RWV\cong\angle FGR$, then $m\angle RWV = m\angle FGR$. Substituting $m\angle RWV$ for $m\angle FGR$ in the equation $m\angle FGR + m\angle STY = 180^{\circ}$, we get $m\angle RWV + m\angle STY=180^{\circ}$.

Answer:

1. Linear - pair postulate (The sum of the measures of two angles that form a linear pair is $180^{\circ}$)
2. Substitution property of equality (Since $m\angle RWV = m\angle FGR$ and $m\angle FGR + m\angle STY = 180^{\circ}$, we substitute $m\angle RWV$ for $m\angle FGR$)