1. **Problem Statement:** We are given a circle with center $P$ and two diameters $\overline{AD}$ and $\overline{BE}$. We need to find the value of $k$ given the angle $\widehat{CAD}$ is composed of two arcs measuring $(20k + 4)^\circ$ and $(33k - 9)^\circ$. 2. **Understanding the Problem:** Since $\overline{AD}$ and $\overline{BE}$ are diameters, they intersect at the center $P$ and divide the circle into four right angles of $90^\circ$ each. 3. **Key Formula:** The measure of an inscribed angle is half the measure of its intercepted arc. Here, $\widehat{CAD}$ is an inscribed angle intercepting the arcs $(20k + 4)^\circ$ and $(33k - 9)^\circ$. 4. **Calculate the total intercepted arc:** $$\text{Arc}_{total} = (20k + 4) + (33k - 9) = 53k - 5$$ 5. **Express the angle $\widehat{CAD}$ in terms of $k$:** $$\widehat{CAD} = \frac{1}{2} \times \text{Arc}_{total} = \frac{1}{2} (53k - 5) = \frac{53k - 5}{2}$$ 6. **Use the fact that $\widehat{CAD}$ is an angle formed by points on the circle:** Since $\overline{AD}$ is a diameter, $\widehat{CAD}$ is a right angle (by Thales' theorem), so: $$\widehat{CAD} = 90^\circ$$ 7. **Set up the equation and solve for $k$:** $$\frac{53k - 5}{2} = 90$$ Multiply both sides by 2: $$53k - 5 = 180$$ Add 5 to both sides: $$53k = 185$$ Divide both sides by 53: $$k = \frac{185}{53}$$ 8. **Final answer:** $$k = \frac{185}{53} \approx 3.49$$ This is the value of $k$ that satisfies the given conditions.