Question

∠uvy ≅ ∠uyv and overleftrightarrow{wx} || overleftrightarrow{vy}. complete the proof that ∠vwx ≅ ∠wxy.

1 ∠uvy ≅ ∠uyv given
2 overleftrightarrow{wx} || overleftrightarrow{vy} given
3 ∠wxy ≅ ∠uyv corresponding angles theorem
4 ∠vwx ≅ ∠uvy
5 ∠vwx ≅ ∠uyv transitive property of congruence
6 ∠vwx ≅ ∠wxy

Explanation:

Step1: Find the reason for $\angle VWX\cong\angle UVY$

Since $\overleftrightarrow{WX}\parallel\overleftrightarrow{VY}$, by the Alternate - Interior Angles Theorem, when a transversal (here \(UV\) is part of the line that can be considered as a transversal for the parallel lines \(\overleftrightarrow{WX}\) and \(\overleftrightarrow{VY}\)), $\angle VWX\cong\angle UVY$.

Step2: Find the reason for $\angle VWX\cong\angle WXY$

We know that \(\angle VWX\cong\angle UVY\) (from step 1) and \(\angle WXY\cong\angle UYV\) (Corresponding Angles Theorem, given \(\overleftrightarrow{WX}\parallel\overleftrightarrow{VY}\)), and \(\angle UVY\cong\angle UYV\) (given). By the Transitive Property of Congruence (if \(a = b\) and \(b = c\), then \(a = c\)), \(\angle VWX\cong\angle WXY\).

Answer:

1. Alternate - Interior Angles Theorem
2. Transitive Property of Congruence (using \(\angle VWX\cong\angle UVY\) from step 4 and \(\angle WXY\cong\angle UYV\) from step 3 and \(\angle UVY\cong\angle UYV\) from statement 1)