1. **State the problem:** A triangle has one angle measuring 85°. The other two angles are in the ratio 2:17. We need to find the measures of these two angles. 2. **Recall the triangle angle sum rule:** The sum of the angles in any triangle is always 180°. So, $$\text{Angle}_1 + \text{Angle}_2 + \text{Angle}_3 = 180^\circ$$ 3. **Set up the equation:** Let the two unknown angles be $2x$ and $17x$ based on the ratio 2:17. We know the first angle is 85°, so: $$85 + 2x + 17x = 180$$ 4. **Simplify the equation:** $$85 + 19x = 180$$ 5. **Isolate $x$:** $$19x = 180 - 85$$ $$19x = 95$$ 6. **Solve for $x$:** $$x = \frac{95}{19}$$ $$x = 5$$ 7. **Find the two angles:** $$2x = 2 \times 5 = 10^\circ$$ $$17x = 17 \times 5 = 85^\circ$$ 8. **Check the sum:** $$85 + 10 + 85 = 180^\circ$$ which confirms the solution is correct. **Final answer:** The two angles measure $10^\circ$ and $85^\circ$.