Triangle Similarity Ca5B67

1. **Problem Statement:** Given that $\overline{NL} \parallel \overline{QO}$ and $\overline{MN} \parallel \overline{PQ}$, prove that $\triangle LMN \sim \triangle OPQ$. 2. **Recall the similarity criteria:** Triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. 3. **Step 1: Identify parallel lines and corresponding angles** Since $\overline{NL} \parallel \overline{QO}$ and $\overline{MN} \parallel \overline{PQ}$, by the Alternate Interior Angles Theorem: - $\angle LNM \cong \angle OQP$ - $\angle NML \cong \angle PQO$ 4. **Step 2: Identify the shared angle** Both triangles share $\angle L$ and $\angle O$ which are vertically aligned on the baseline $\overline{LM}$ and $\overline{OP}$ respectively. Thus, $\angle L \cong \angle O$. 5. **Step 3: Conclude angle-angle similarity** Since two pairs of corresponding angles are congruent: - $\angle LNM \cong \angle OQP$ - $\angle NML \cong \angle PQO$ And the third pair $\angle L \cong \angle O$ by shared baseline, we have $\triangle LMN \sim \triangle OPQ$ by the AA (Angle-Angle) similarity criterion. 6. **Final conclusion:** $\triangle LMN \sim \triangle OPQ$ is proven by AA similarity due to parallel lines creating congruent alternate interior angles and the shared baseline angle. **Answer:** $\triangle LMN \sim \triangle OPQ$