1. **Problem 4: Are triangles MNO and XYZ similar?** Given sides: - Triangle MNO: 5 cm, 10 cm, 15 cm - Triangle XYZ: 3 cm, 6 cm, 9 cm 2. **Formula and rule:** Triangles are similar if their corresponding sides are in proportion, i.e., the ratios of corresponding sides are equal. 3. **Calculate ratios of corresponding sides:** $$\frac{5}{3} = \frac{10}{6} = \frac{15}{9}$$ Simplify each ratio: $$\frac{5}{3} \approx 1.6667$$ $$\frac{10}{6} = \frac{\cancel{10}}{\cancel{6}} = \frac{5}{3} \approx 1.6667$$ $$\frac{15}{9} = \frac{\cancel{15}}{\cancel{9}} = \frac{5}{3} \approx 1.6667$$ 4. **Conclusion:** Since all three ratios are equal, the triangles MNO and XYZ are similar. 1. **Problem 5: Find angles 6 and 4 given angle 2 = 112°** (a) Angle 6 is corresponding to angle 2, so: $$\text{Angle 6} = 112^\circ$$ (b) Angle 4 is vertical to angle 2, so vertical angles are equal: $$\text{Angle 4} = 112^\circ$$ 1. **Problem 6: Transform triangle DEF and find final coordinates** Given vertices: $$D(-3,2), E(1,2), F(-1,5)$$ Step 1: Rotate 90° clockwise about the origin. Rotation rule for 90° clockwise: $$(x,y) \to (y, -x)$$ Apply to each point: $$D' = (2, 3)$$ $$E' = (2, -1)$$ $$F' = (5, 1)$$ Step 2: Translate 2 units up. Translation rule: $$(x,y) \to (x, y+2)$$ Apply to each rotated point: $$D'' = (2, 3+2) = (2, 5)$$ $$E'' = (2, -1+2) = (2, 1)$$ $$F'' = (5, 1+2) = (5, 3)$$ **Final coordinates:** $$D''(2,5), E''(2,1), F''(5,3)$$ **Congruence:** Rotation and translation are rigid transformations that preserve distance and angle measures, so the final image is congruent to the original triangle DEF.