Question

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

statement
1 ∠uvy ≅ ∠uyv
2 overleftrightarrow{wx} || overleftrightarrow{vy}
3 ∠wxy ≅ ∠uyv
4 ∠vwx ≅ ∠uvy
5 ∠vwx ≅ ∠uyv
6 ∠vwx ≅ ∠wxy

reason
given
given
corresponding angles theorem
transitive property of congruence

Explanation:

Step1: Identify the reason for ∠VWX ≅ ∠UVY

Since \( \overleftrightarrow{WX}\parallel\overleftrightarrow{VY}\), by the Alternate Interior Angles Theorem, \( \angle VWX\cong\angle UVY\).

Step2: Justify ∠VWX ≅ ∠WXY

We know \( \angle WXY\cong\angle UYV\) (Corresponding Angles Theorem) and \( \angle UVY\cong\angle UYV\) (Given). Also, \( \angle VWX\cong\angle UVY\) (Alternate Interior Angles Theorem). By the Transitive Property of Congruence, if \( a = b\) and \( b = c\), then \( a=c\). Here, since \( \angle VWX\cong\angle UVY\) and \( \angle WXY\cong\angle UYV\) and \( \angle UVY\cong\angle UYV\), we can say \( \angle VWX\cong\angle WXY\) by substitution (Transitive Property of Congruence).

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 step 1)