Question

given: $overline{xy} parallel overline{zw}$
prove: $\triangle xyv sim \triangle zwv$

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 dropdown and angle dropdown are alternate interior angles. since $overline{xy}$ is parallel to $overline{zw}$, we know that these angles are dropdown. we also know that angle $xvy$ and angle $zwv$ are dropdown angles, and thus congruent. we can conclude that $\triangle xyv sim \triangle zwv$ using the dropdown similarity theorem. (dropdown options include sss, aa, hl.)

Explanation:

Step1: 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 first two blanks: \( \angle X \) and \( \angle Z \).

Step2: Property of Alternate Interior Angles

Since \( \overline{XY} \parallel \overline{ZW} \), alternate interior angles are congruent. So third blank: congruent.

Step3: Identify Vertical Angles

\( \angle XVY \) and \( \angle ZVW \) are vertical angles (formed by intersection of \( \overline{XZ} \) and \( \overline{YW} \)), so they are congruent. Fourth blank: vertical.

Step4: Similarity Theorem

We have two pairs of congruent angles (\( \angle X \cong \angle Z \), \( \angle XVY \cong \angle ZVW \)), so by AA (Angle - Angle) similarity theorem, \( \triangle XYV \sim \triangle ZWV \). Fifth blank: AA.

Answer:

First drop - down: \( \boldsymbol{\angle X} \); Second drop - down: \( \boldsymbol{\angle Z} \); Third drop - down: \( \boldsymbol{\text{congruent}} \); Fourth drop - down: \( \boldsymbol{\text{vertical}} \); Fifth drop - down: \( \boldsymbol{\text{AA}} \)