Question

complete the proof that $m\angle wvy = m\angle u + m\angle y$ without using the triangle angle sum theorem.
(image of a triangle and angles with a table of statements and reasons)

1. $\overline{vx} \parallel \overline{uy}$: given
2. $\angle u \cong \angle wvx$: corresponding angles theorem
3. $\angle y \cong \angle xvy$: dropdown
4. $m\angle wvy = m\angle wvx + m\angle xvy$: dropdown
5. $m\angle wvy = m\angle u + m\angle y$: dropdown

Explanation:

Step1: Analyze Step 3 Reason

Since \( \overline{VX} \parallel \overline{UY} \), \( \angle Y \) and \( \angle XVY \) are alternate interior angles. So the reason for \( \angle Y \cong \angle XVY \) is the Alternate Interior Angles Theorem.

Step2: Analyze Step 4 Reason

\( \angle WVY \) is composed of \( \angle WVX \) and \( \angle XVY \), so by the Angle Addition Postulate, \( m\angle WVY = m\angle WVX + m\angle XVY \).

Step3: Analyze Step 5 Reason

From Step 2, \( \angle U \cong \angle WVX \) (so \( m\angle U = m\angle WVX \)) and from Step 3, \( \angle Y \cong \angle XVY \) (so \( m\angle Y = m\angle XVY \)). Substituting these into Step 4's equation gives \( m\angle WVY = m\angle U + m\angle Y \) by Substitution Property.

Answer:

• Step 3 Reason: Alternate Interior Angles Theorem
• Step 4 Reason: Angle Addition Postulate
• Step 5 Reason: Substitution Property (using \( m\angle U = m\angle WVX \) and \( m\angle Y = m\angle XVY \))