Angle Congruence 988Ffd

1. **State the problem:** We are given that $\overline{UV} \parallel \overline{ST}$, $\overline{QR} \parallel \overline{ST}$, and $\angle 4 \cong \angle 3$. We need to prove that $\angle 8 \cong \angle 1$. 2. **Recall important properties:** - When two lines are parallel, corresponding angles formed by a transversal are congruent. - Alternate interior angles formed by a transversal with parallel lines are congruent. 3. **Use given information:** - From $\overline{UV} \parallel \overline{ST}$ and the transversal that creates $\angle 4$ and $\angle 3$, we know $\angle 4 \cong \angle 3$ (given). 4. **Identify transversal and angles:** - Since $\overline{QR} \parallel \overline{ST}$, and $\overline{UV} \parallel \overline{ST}$, $\overline{UV}$ and $\overline{QR}$ are both parallel to $\overline{ST}$. 5. **Use transitive property of parallel lines:** - Because $\overline{UV} \parallel \overline{ST}$ and $\overline{QR} \parallel \overline{ST}$, $\overline{UV} \parallel \overline{QR}$. 6. **Identify transversal creating $\angle 8$ and $\angle 1$:** - The transversal crossing $\overline{UV}$ and $\overline{QR}$ creates $\angle 8$ and $\angle 1$. 7. **Apply corresponding angles postulate:** - Since $\overline{UV} \parallel \overline{QR}$, $\angle 8 \cong \angle 1$ because they are corresponding angles. **Final conclusion:** $\angle 8 \cong \angle 1$ is proven by the properties of parallel lines and corresponding angles.