Question

complete the proof that m∠u + m∠y = m∠wvy without using the triangle angle sum theorem.

image of a geometric diagram with points w, v, x, u, y and a table with statements and reasons

statement | reason
--- | ---

1. ( overline{uy} parallel overline{vx} ) | given
2. ( angle u cong angle wvx ) | alternate interior angles theorem
3. ( angle y cong angle xvy ) | dropdown for reason
4. ( mangle wvx + mangle xvy = mangle wvy ) | dropdown for reason
5. ( mangle u + mangle y = mangle wvy ) | dropdown for reason

(reasons dropdown includes: alternate exterior angles theorem, alternate interior angles theorem, converse of alternate exterior angles theorem, converse of alternate interior angles theorem, converse of corresponding angles theorem, converse of same - side exterior angles theorem)

Explanation:

Step1: Analyze Angle Relationships

We know \( \overline{UY} \parallel \overline{VX} \) (given). For \( \angle Y \cong \angle XVY \), since \( \overline{UY} \parallel \overline{VX} \) and \( \overline{VY} \) is a transversal, the Alternate Interior Angles Theorem applies (alternate interior angles are congruent when lines are parallel). So the reason for step 3 is Alternate Interior Angles Theorem.

Step2: Analyze Step 4

The statement \( m\angle WVX + m\angle XVY = m\angle WVY \) is the Angle Addition Postulate (the sum of two adjacent angles is equal to the measure of the angle they form together). Wait, but looking at the options, maybe we misread. Wait, step 4: \( \angle WVX \) and \( \angle XVY \) are adjacent angles forming \( \angle WVY \), so the reason is Angle Addition Postulate, but in the dropdown, maybe a typo? Wait, no, the problem is about the proof. Then step 5: since \( \angle U \cong \angle WVX \) (step 2) and \( \angle Y \cong \angle XVY \) (step 3), by Substitution Property (replacing \( \angle WVX \) with \( \angle U \) and \( \angle XVY \) with \( \angle Y \) in step 4's equation), we get \( m\angle U + m\angle Y = m\angle WVY \). But the dropdown for step 3: since \( \overline{UY} \parallel \overline{VX} \) and transversal \( VY \), alternate interior angles \( \angle Y \) and \( \angle XVY \) are congruent, so reason is Alternate Interior Angles Theorem.

Step3: Confirm Step 3 Reason

From the diagram, \( \overline{UY} \parallel \overline{VX} \), transversal \( VY \), so \( \angle Y \) and \( \angle XVY \) are alternate interior angles. Thus, the reason for step 3 is Alternate Interior Angles Theorem.

Step4: Confirm Step 5 Reason

Step 2: \( \angle U \cong \angle WVX \) (Alternate Interior Angles, transversal \( VU \)? Wait, no, \( \overline{UY} \parallel \overline{VX} \), transversal \( VU \)? Wait, maybe \( \overline{VU} \) is parallel? No, the given is \( \overline{UY} \parallel \overline{VX} \). So step 2: \( \angle U \) and \( \angle WVX \) are alternate interior angles (transversal \( VU \)? Maybe. Then step 3: \( \angle Y \) and \( \angle XVY \) (transversal \( VY \)). Then step 4: angle addition (adjacent angles). Step 5: substitute \( \angle U \) for \( \angle WVX \) and \( \angle Y \) for \( \angle XVY \), so Substitution Property. But in the dropdown, the options include Alternate Interior Angles Theorem for step 3, which is correct.

Wait, the problem is to complete the proof. Let's re-express:

1. \( \overline{UY} \parallel \overline{VX} \) (Given)
2. \( \angle U \cong \angle WVX \) (Alternate Interior Angles Theorem, transversal \( VU \))
3. \( \angle Y \cong \angle XVY \) (Alternate Interior Angles Theorem, transversal \( VY \))
4. \( m\angle WVX + m\angle XVY = m\angle WVY \) (Angle Addition Postulate)
5. \( m\angle U + m\angle Y = m\angle WVY \) (Substitution Property, replacing \( \angle WVX \) with \( \angle U \) and \( \angle XVY \) with \( \angle Y \) from steps 2 and 3)

But the dropdown for step 3: the reason is Alternate Interior Angles Theorem (as \( \overline{UY} \parallel \overline{VX} \), transversal \( VY \), so alternate interior angles \( \angle Y \) and \( \angle XVY \) are congruent).

For step 5, the reason would be Substitution (since \( \angle U = \angle WVX \) and \( \angle Y = \angle XVY \), so substitute into step 4's equation). But in the dropdown, maybe the options have "Substitution" or related, but the user's dropdown for step 3: the correct reason is Alternate Interior Angles Theorem.

Answer:

For step 3, the reason is Alternate Interior Angles Theorem. For step 5, by substituting \( \angle U \) for \( \angle WVX \) and \( \angle Y \) for \( \angle XVY \) (from steps 2 and 3) into step 4's equation, we get \( m\angle U + m\angle Y = m\angle WVY \), so the proof is complete. The key reasons are Alternate Interior Angles Theorem for steps 2 and 3, Angle Addition Postulate for step 4, and Substitution for step 5. But focusing on the dropdown for step 3, the answer is Alternate Interior Angles Theorem.