1. **Problem Statement:** We are given a circle with points A, B, C, D on its circumference and angles marked as follows: $\angle A = 55^\circ$ (right angle), $\angle B = 39^\circ$, $\angle E = 35^\circ$, and $\angle C = x^\circ$. Segments CA and AB are equal in length. 2. **Key Information and Formula:** Since CA = AB, triangle CAB is isosceles with $\angle C = \angle B$ or $\angle C = x$ and $\angle B = 39^\circ$. Also, $\angle A = 90^\circ$ (right angle). 3. **Using Triangle Angle Sum:** The sum of angles in triangle CAB is $180^\circ$. So, $$x + 39 + 55 = 180$$ 4. **Calculate $x$:** $$x = 180 - 39 - 55 = 86$$ 5. **Conclusion:** The value of $x$ is $86^\circ$.