Question

list the pairs of congruent angles and the extended proportion that relates the corresponding sides for the similar polygons.
uvwx ~ mnpq

list the pairs of congruent angles.
∠u ≅ ∠□
∠v ≅ ∠□
∠w ≅ ∠□
∠x ≅ ∠□

Explanation:

Step1: Recall Similar Polygons Property

In similar polygons, corresponding angles are congruent. The notation \( UVWX \sim MNPQ \) means the order of the letters gives the corresponding vertices. So, \( U \) corresponds to \( M \), \( V \) corresponds to \( N \), \( W \) corresponds to \( P \), and \( X \) corresponds to \( Q \).

Step2: Identify Congruent Angles

• For \( \angle U \), the corresponding angle is \( \angle M \) (since \( U \) corresponds to \( M \) in the similarity \( UVWX \sim MNPQ \)).
• For \( \angle V \), the corresponding angle is \( \angle N \) (since \( V \) corresponds to \( N \)).
• For \( \angle W \), the corresponding angle is \( \angle P \) (since \( W \) corresponds to \( P \)).
• For \( \angle X \), the corresponding angle is \( \angle Q \) (since \( X \) corresponds to \( Q \)).

Answer:

\( \angle U \cong \angle M \)
\( \angle V \cong \angle N \)
\( \angle W \cong \angle P \)
\( \angle X \cong \angle Q \)

(For the blanks in the original problem:
\( \angle U \cong \angle \boldsymbol{M} \)
\( \angle V \cong \angle \boldsymbol{N} \)
\( \angle W \cong \angle \boldsymbol{P} \)
\( \angle X \cong \angle \boldsymbol{Q} \))