1. **State the problem:** We are given two triangles with parallel sides BC and DE, and a proportion involving side lengths: $$\frac{16}{?} = \frac{?}{21}$$. We need to find the unknown side lengths represented by "?". 2. **Formula and theorem:** When two lines are parallel in triangles, corresponding sides are proportional. This is the Angle Proportionality Theorem, which states: $$\frac{AB}{BD} = \frac{AE}{DE} = \frac{AC}{CE}$$ 3. **Set variables:** Let the unknown sides be $x$. The proportion is: $$\frac{16}{x} = \frac{x}{21}$$ 4. **Solve the proportion:** Cross-multiply: $$16 \times 21 = x \times x$$ $$336 = x^2$$ 5. **Find $x$:** Take the square root of both sides: $$x = \sqrt{336}$$ 6. **Simplify $\sqrt{336}$:** $$336 = 16 \times 21$$ $$\sqrt{336} = \sqrt{16 \times 21} = \sqrt{16} \times \sqrt{21} = 4\sqrt{21}$$ 7. **Final answer:** $$x = 4\sqrt{21}$$ This means the unknown side lengths are $4\sqrt{21}$ units.