1. **State the problem:** We have a right isosceles triangle RST with right angle at T, sides RT = ST = 23 units, and base RS = t. We need to find the measures of angles $\angle R$ and $\angle S$. 2. **Recall properties:** In a right triangle, the sum of angles is 90° + 90° = 180°. Since $\angle T = 90^\circ$, the other two angles $\angle R$ and $\angle S$ must sum to 90°. 3. **Isosceles right triangle:** Since RT = ST, the triangle is isosceles with legs equal, so $\angle R = \angle S$. 4. **Calculate angles:** Let $\angle R = \angle S = x$. Then: $$x + x = 90^\circ$$ $$2x = 90^\circ$$ $$x = \frac{90^\circ}{2} = 45^\circ$$ 5. **Final answer:** $$m\angle R = 45^\circ$$ $$m\angle S = 45^\circ$$ Thus, both angles R and S measure 45 degrees each.