1. **State the problem:** Given that $E$ is the midpoint of segments $GN$ and $WQ$, prove that $WN \cong GQ$. 2. **Understand the given information:** Since $E$ is the midpoint of $GN$ and $WQ$, it means: - $GE = EN$ - $WE = EQ$ 3. **Identify the angles:** The proof mentions the following angle congruences: - $\angle GEQ \cong \angle NEW$ (Vertical angles are congruent) - $\angle G \cong \angle N$ (Alternate interior angles) - $\angle W \cong \angle Q$ (Alternate interior angles) 4. **Use triangle congruence criteria:** Consider triangles $WNE$ and $GQE$. - Side $WE = EQ$ (since $E$ is midpoint of $WQ$) - Side $GE = EN$ (since $E$ is midpoint of $GN$) - $\angle GEQ \cong \angle NEW$ (vertical angles) 5. **Apply the Side-Angle-Side (SAS) postulate:** - In $\triangle WNE$ and $\triangle GQE$, two sides and the included angle are congruent. 6. **Conclude:** By SAS, $WN \cong GQ$. **Final answer:** $WN \cong GQ$ is proven by the SAS postulate using the given midpoints and angle congruences.