1. **Problem Statement:** Given multiple triangles with labeled sides and vertices, find the unknown side lengths using similarity properties. 2. **Similarity Rule:** Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. 3. **Step 1: Analyze Triangle CBA** - Given sides: CB = 6, AE = y, DE = 7(y-1), and variable x on CB. - Since BẮC = DBC, triangles involving these points are similar. 4. **Step 2: Set up proportions for similar triangles:** - For triangles with sides proportional, write: $$\frac{CB}{DE} = \frac{AE}{x}$$ - Substitute known values: $$\frac{6}{7(y-1)} = \frac{y}{x}$$ 5. **Step 3: Solve for x or y as needed:** - Cross multiply: $$6x = y \times 7(y-1)$$ - Simplify right side: $$6x = 7y(y-1) = 7y^2 - 7y$$ 6. **Step 4: Analyze Triangle ABC with sides m, n, 3, 4, 2** - Use similarity to set ratios: $$\frac{m}{3} = \frac{n}{4} = \frac{?}{2}$$ - Solve for unknowns by cross multiplication. 7. **Step 5: Triangle PQR with sides 2, 3, 5 and segments m, 3, 2, 5** - Use similarity and segment addition to find m. 8. **Step 6: Triangle XYZ with sides XY=3, WX=5, WZ=4, and right angle at Y** - Use Pythagoras theorem for right triangle: $$WX^2 = XY^2 + YZ^2$$ - Substitute: $$5^2 = 3^2 + u^2$$ - Calculate: $$25 = 9 + u^2$$ - Solve for u: $$u^2 = 16 \Rightarrow u = 4$$ **Final answers:** - From step 3: $6x = 7y^2 - 7y$ - From step 8: $u = 4$ Further values depend on additional data or equations not fully provided.