1. **Problem:** Determine if two triangular signs with sides 20 cm, 30 cm, 40 cm and 40 cm, 60 cm, 80 cm are similar. 2. **Formula and rule:** Triangles are similar if their corresponding sides are proportional. 3. **Check ratios:** $$\frac{20}{40} = \frac{1}{2}, \quad \frac{30}{60} = \frac{1}{2}, \quad \frac{40}{80} = \frac{1}{2}$$ 4. **Explanation:** All corresponding side lengths have the same ratio $\frac{1}{2}$, so the triangles are similar. 2. **Problem:** Find the missing side $z$ in a right triangle with sides 9, $z$, 12, and hypotenuse 15. 3. **Formula:** Use the Pythagorean theorem for right triangles: $$a^2 + b^2 = c^2$$ where $c$ is the hypotenuse. 4. **Apply values:** $$9^2 + z^2 = 15^2$$ $$81 + z^2 = 225$$ 5. **Solve for $z^2$:** $$z^2 = 225 - 81$$ $$z^2 = 144$$ 6. **Find $z$:** $$z = \sqrt{144} = 12$$ 7. **Conclusion:** The missing side $z$ is 12.