1. **Stating the problem:** We have two triangles, ABC and CDE, which are similar by the given angle conditions: $\angle B = \angle D$ and $\angle A = \angle E$. Given side lengths are $AB = 12$ cm, $AC = 15$ cm, $BC = 18$ cm, and $DE = 10$ cm. We need to find the unknown sides of triangle CDE using similarity. 2. **Formula and rules:** For similar triangles, corresponding sides are proportional. This means: $$\frac{AB}{CD} = \frac{BC}{DE} = \frac{AC}{CE}$$ where sides correspond as follows: $AB \leftrightarrow CD$, $BC \leftrightarrow DE$, and $AC \leftrightarrow CE$. 3. **Find side $CD$:** Using the proportion between $BC$ and $DE$: $$\frac{BC}{DE} = \frac{18}{10} = 1.8$$ Since $\frac{AB}{CD} = 1.8$, we have: $$\frac{12}{CD} = 1.8$$ Multiply both sides by $CD$: $$12 = 1.8 \times CD$$ Divide both sides by 1.8: $$CD = \frac{12}{1.8}$$ Show cancellation: $$CD = \frac{\cancel{12}}{\cancel{1.8}} = 6.666... \approx 6.67\text{ cm}$$ 4. **Find side $CE$:** Using the proportion between $AC$ and $CE$: $$\frac{AC}{CE} = 1.8$$ So: $$\frac{15}{CE} = 1.8$$ Multiply both sides by $CE$: $$15 = 1.8 \times CE$$ Divide both sides by 1.8: $$CE = \frac{15}{1.8}$$ Show cancellation: $$CE = \frac{\cancel{15}}{\cancel{1.8}} = 8.333... \approx 8.33\text{ cm}$$ 5. **Summary:** The sides of triangle CDE are: - $CD \approx 6.67$ cm - $DE = 10$ cm (given) - $CE \approx 8.33$ cm These lengths maintain the similarity ratio of 1.8 between triangles ABC and CDE.