1. **State the problem:** We need to find the measure of angle $\angle AB$ in triangle $ABC$ with a right angle at $D$ and given angles $71^\circ$ at $A$ and $71^\circ$ at $C$. 2. **Given information:** - $\angle A = 71^\circ$ - $\angle D = 90^\circ$ - $\angle C = 71^\circ$ 3. **Check the sum of angles in triangle $ABC$:** The sum of interior angles in any triangle is $180^\circ$. 4. **Calculate $\angle B$:** $$71^\circ + 90^\circ + B = 180^\circ$$ 5. **Solve for $B$:** $$B = 180^\circ - 71^\circ - 90^\circ$$ $$B = 19^\circ$$ 6. **Interpretation:** The calculation is correct. The measure of $\angle AB$ (which corresponds to $B$) is $19^\circ$. **Final answer:** $$\boxed{19^\circ}$$