1. **State the problem:** We have a triangle with one angle measuring 30° and the other two angles in the ratio 13:17. 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. **Assign variables:** Let the two unknown angles be $13x$ and $17x$ based on the given ratio. 4. **Set up the equation:** Using the sum of angles, $$30 + 13x + 17x = 180$$ 5. **Simplify the equation:** $$30 + 30x = 180$$ 6. **Isolate $x$:** $$30x = 180 - 30$$ $$30x = 150$$ $$x = \frac{150}{30}$$ $$x = 5$$ 7. **Find the angles:** $$13x = 13 \times 5 = 65^\circ$$ $$17x = 17 \times 5 = 85^\circ$$ 8. **Check the sum:** $$30 + 65 + 85 = 180^\circ$$ which confirms the solution is correct. **Final answer:** The two angles measure $65^\circ$ and $85^\circ$.