Question

$$\overleftrightarrow{eg}\parallel\overleftrightarrow{vx}$$ and $$\overleftrightarrow{su}\parallel\overleftrightarrow{eg}$$. complete the proof that $$m\angle rts + m\angle vwy = 180^{\circ}$$.

Explanation:

Step1: Use the transitive property of parallel lines

Since \(\overleftrightarrow{EG}\parallel\overleftrightarrow{VX}\) (given) and \(\overleftrightarrow{SU}\parallel\overleftrightarrow{EG}\) (given), by the transitive property of parallel lines (\(a\parallel b\) and \(b\parallel c\) implies \(a\parallel c\)), we have \(\overleftrightarrow{SU}\parallel\overleftrightarrow{VX}\).

Step2: Use the substitution property

We know that \(\angle RTS\cong\angle EFR\) (Corresponding Angles Theorem). This means \(m\angle RTS = m\angle EFR\).
We also know that \(m\angle EFR+m\angle VWY = 180^{\circ}\) (Same - Side Interior Angles Theorem for \(\overleftrightarrow{EG}\parallel\overleftrightarrow{VX}\)).
Substitute \(m\angle EFR\) with \(m\angle RTS\) in the equation \(m\angle EFR+m\angle VWY = 180^{\circ}\).

Answer:

Substitution Property (because \(m\angle RTS\) is substituted for \(m\angle EFR\) in the equation \(m\angle EFR + m\angle VWY=180^{\circ}\))