1. **Problem Statement:** We are given a triangle with points A, B, C, and D. Segments AC = 5.5, AB = 5.3, BD = 4.1, and angles $\angle DAC = \angle BAD = \theta$. We need to find the length of segment CD. 2. **Understanding the problem:** Since $\angle DAC = \angle BAD$, point D lies on the angle bisector of $\angle BAC$. By the Angle Bisector Theorem, the angle bisector divides the opposite side into segments proportional to the adjacent sides. 3. **Angle Bisector Theorem:** If D lies on BC such that $\angle DAC = \angle BAD$, then $$\frac{CD}{DB} = \frac{AC}{AB}$$ 4. **Substitute known values:** $$\frac{CD}{4.1} = \frac{5.5}{5.3}$$ 5. **Solve for CD:** $$CD = 4.1 \times \frac{5.5}{5.3}$$ 6. **Calculate:** $$CD = 4.1 \times 1.0377 = 4.2525$$ 7. **Round to one decimal place:** $$CD \approx 4.3$$ **Final answer:** The length of segment CD is approximately **4.3**.