1. **Problem statement:** Given a right triangle KLM with angles at K and L both 45°, and side KM = 20, find the length of side LM. 2. **Understanding the problem:** Since angles K and L are both 45°, triangle KLM is an isosceles right triangle (45°-45°-90° triangle). 3. **Formula for 45°-45°-90° triangle:** The sides are in the ratio $1:1:\sqrt{2}$, where the hypotenuse is $\sqrt{2}$ times each leg. 4. **Identify sides:** Here, KM is the hypotenuse, so $KM = 20$. 5. **Find leg length LM:** Using the ratio, each leg is $\frac{KM}{\sqrt{2}}$. 6. **Calculate LM:** $$ LM = \frac{20}{\sqrt{2}} = \frac{20}{\cancel{\sqrt{2}}} \times \frac{\cancel{\sqrt{2}}}{\sqrt{2}} = \frac{20\sqrt{2}}{2} = 10\sqrt{2} $$ 7. **Answer:** The length of LM is $10\sqrt{2}$.