1. **State the problem:** We need to find the measure of angle $\angle ABD$ in the quadrilateral $ABD$ where $\angle BAD = (7x + 1)^\circ$, $\angle ABD = (2x + 36)^\circ$, and sides $AD$ and $BD$ are both 54 units with right angles at $A$ and $D$. 2. **Analyze the figure:** Since $\angle A$ and $\angle D$ are right angles, each measures $90^\circ$. 3. **Use the angle sum property of quadrilaterals:** The sum of interior angles in any quadrilateral is $360^\circ$. 4. **Set up the equation:** $$\angle A + \angle B + \angle D + \angle C = 360^\circ$$ Here, $\angle A = 90^\circ$, $\angle D = 90^\circ$, $\angle B = (7x + 1)^\circ + (2x + 36)^\circ$ (since $\angle BAD$ and $\angle ABD$ are parts of $\angle B$), and $\angle C$ is the remaining angle. 5. **Sum the parts of $\angle B$:** $$\angle B = (7x + 1) + (2x + 36) = 9x + 37$$ 6. **Write the full angle sum equation:** $$90 + (9x + 37) + 90 + \angle C = 360$$ 7. **Simplify:** $$180 + 9x + 37 + \angle C = 360$$ $$9x + \angle C + 217 = 360$$ 8. **Isolate $\angle C$:** $$\angle C = 360 - 217 - 9x = 143 - 9x$$ 9. **Use the fact that $AD = BD = 54$:** Triangle $ABD$ is isosceles with sides $AD = BD$, so angles opposite these sides are equal. 10. **Therefore, $\angle BAD = \angle ABD$:** $$7x + 1 = 2x + 36$$ 11. **Solve for $x$:** $$7x + 1 = 2x + 36$$ $$7x - 2x = 36 - 1$$ $$5x = 35$$ $$x = 7$$ 12. **Find $\angle ABD$:** $$\angle ABD = 2x + 36 = 2(7) + 36 = 14 + 36 = 50^\circ$$ **Final answer:** $$\boxed{50^\circ}$$