Question

given: $overline{xy} parallel overline{zw}$
prove: $\triangle xyw sim \triangle zwv$
use the drop-down menus to complete the paragraph proof
we are given that $overline{xy}$ is parallel to $overline{zw}$. if $overline{xz}$ is a transversal that intercepts $overline{xy}$ and $overline{zw}$, angle $\boldsymbol{square}$ and angle $\boldsymbol{square}$ are alternate interior angles. since $overline{xy}$ is parallel to $overline{zw}$, we know that these angles are $\boldsymbol{square}$. we also know that angle xvy and angle zvw are $\boldsymbol{square}$ angles, and thus congruent. therefore, $\triangle xyw sim \triangle zwv$ using the $\boldsymbol{square}$ theorem.
drop-down options: corresponding, vertical, supplementary

Explanation:

Step 1: Identify Alternate Interior Angles

When transversal \( \overline{XZ} \) intersects parallel lines \( \overline{XY} \) and \( \overline{ZW} \), the alternate interior angles are \( \angle X \) (at \( X \) in \( \triangle XYV \)) and \( \angle Z \) (at \( Z \) in \( \triangle ZWV \)). So the first two blanks are \( \angle X \) and \( \angle Z \).

Step 2: Property of Alternate Interior Angles

By the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, alternate interior angles are congruent. So the third blank is "congruent".

Step 3: Identify Angle Relationship (Vertical Angles)

\( \angle XVY \) and \( \angle ZVW \) are vertical angles (formed by the intersection of \( \overline{XZ} \) and \( \overline{WY} \)). Vertical angles are congruent. So the fourth blank is "vertical".

Step 4: Similarity Criterion

To prove \( \triangle XYV \sim \triangle ZWV \), we use the AA (Angle - Angle) Similarity Theorem, which states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. We have \( \angle X \cong \angle Z \) (alternate interior angles) and \( \angle XVY \cong \angle ZVW \) (vertical angles), so by AA similarity, the triangles are similar. The last blank (the theorem) is "AA (Angle - Angle)" (though the dropdown had "corresponding" which might be a typo, but following the proof logic, vertical angles and alternate interior angles give AA).

Answer:

1. First angle blank: \( \boldsymbol{\angle X} \)
2. Second angle blank: \( \boldsymbol{\angle Z} \)
3. Third blank: \( \boldsymbol{\text{congruent}} \)
4. Fourth blank: \( \boldsymbol{\text{vertical}} \)
5. Theorem blank: \( \boldsymbol{\text{AA (Angle - Angle)}} \) (Note: If the dropdown strictly has "corresponding" as an option, there might be a mis - labeling, but the correct relationship for the angles here leads to AA similarity. If we follow the dropdown options given for the fourth blank, "vertical" is correct for \( \angle XVY \) and \( \angle ZVW \), and the similarity theorem is AA which is related to corresponding angles being congruent in the similarity context.)