Question

\\( \angle u v y \cong \angle u y v \\) and \\( \overleftrightarrow{w x} \\| \overleftrightarrow{v y} \\). complete the proof that \\( \angle v w x \cong \angle w x y \\).

1
\\( \angle u v y \cong \angle u y v \\)
given
2
\\( \overleftrightarrow{w x} \\| \overleftrightarrow{v y} \\)
given
3
\\( \angle w x y \cong \angle u y v \\)
corresponding angles theorem
4
\\( \angle v w x \cong \angle u v y \\)

5
\\( \angle v w x \cong \angle u y v \\)
transitive property of congruence
6
\\( \angle v w x \cong \angle w x y \\)

Explanation:

Step1: Identify the reason for ∠VWX ≅ ∠UVY

Since \( \overleftrightarrow{WX}\parallel\overleftrightarrow{VY}\), and we are looking at the relationship between \(∠VWX\) and \(∠UVY\), we use the Alternate - Interior Angles Theorem. When two parallel lines (\(\overleftrightarrow{WX}\) and \(\overleftrightarrow{VY}\)) are cut by a transversal (\(UV\)), alternate - interior angles are congruent.

Step2: Identify the reason for ∠VWX ≅ ∠WXY

We know that \(∠VWX≅∠UYV\) (from step 4 and the transitive property in step 5 uses \(∠UVY ≅ ∠UYV\) (given in statement 1) and \(∠VWX ≅ ∠UVY\)). Also, \(∠WXY≅∠UYV\) (statement 3). By the Transitive Property of Congruence (if \(a = b\) and \(b = c\), then \(a = c\)), since \(∠VWX≅∠UYV\) and \(∠WXY≅∠UYV\), we can conclude \(∠VWX≅∠WXY\). The reason for statement 6 is also the Transitive Property of Congruence (using \(∠VWX≅∠UYV\) and \(∠WXY≅∠UYV\))

Answer:

• For statement 4: Alternate - Interior Angles Theorem
• For statement 6: Transitive Property of Congruence