1. **State the problem:** Given that quadrilaterals LMNO and PQRS are similar, we need to complete the proportions and congruence statements and find the value of $x$. 2. **Similarity of polygons:** When two polygons are similar, their corresponding angles are congruent and their corresponding sides are proportional. 3. **Identify corresponding angles:** Since LMNO ~ PQRS, the corresponding angles are: - $\angle L \leftrightarrow \angle P$ - $\angle M \leftrightarrow \angle Q$ - $\angle N \leftrightarrow \angle R$ - $\angle O \leftrightarrow \angle S$ 4. **Answer parts a and b:** - a. $\angle P = \angle L$ - b. $\angle M = \angle Q$ 5. **Set up side length proportions:** Corresponding sides are: - $LM \leftrightarrow PQ$ - $MN \leftrightarrow QR$ - $NO \leftrightarrow RS$ - $OL \leftrightarrow SP$ Given side lengths: - $LM = 36$ - $PQ = 24$ - $MN = x - 2$ - $QR = 18$ 6. **Write the proportion for part c:** $$\frac{MN}{RQ} = \frac{LM}{PQ}$$ 7. **Substitute known values:** $$\frac{x - 2}{18} = \frac{36}{24}$$ 8. **Simplify the right side:** $$\frac{36}{24} = \frac{3}{2}$$ 9. **Solve for $x$:** Multiply both sides by 18: $$x - 2 = 18 \times \frac{3}{2}$$ $$x - 2 = 27$$ Add 2 to both sides: $$x = 27 + 2$$ $$x = 29$$ **Final answers:** - a. $\angle P = \angle L$ - b. $\angle M = \angle Q$ - c. $\frac{MN}{RQ} = \frac{LM}{PQ}$ - d. $x = 29$