1. **Problem Statement:** Find the measure of angle $\angle BCD$ given the angles $\angle D = 85^\circ$, $\angle A = 20^\circ$, $\angle C = 50^\circ$, and $\angle B = 100^\circ$ in a circle with points A, B, C, D, and E. 2. **Understanding the problem:** The points lie on a circle, and the angles given are likely inscribed angles or angles formed by chords intersecting inside the circle. We want to find $\angle BCD$, which is an angle at point C formed by points B and D. 3. **Key rule:** The measure of an inscribed angle is half the measure of its intercepted arc. Also, angles around a point sum to $360^\circ$, and angles in a triangle sum to $180^\circ$. 4. **Step-by-step solution:** - Given $\angle C = 50^\circ$ and $\angle B = 100^\circ$, and $\angle D = 85^\circ$, these angles likely correspond to angles at points C, B, and D respectively. - Since $\angle B = 100^\circ$ and $\angle D = 85^\circ$, and $\angle A = 20^\circ$, the sum of these angles is $100^\circ + 85^\circ + 20^\circ = 205^\circ$. - The remaining angle at point C is $50^\circ$ as given. - To find $\angle BCD$, note that it is the angle at C formed by points B and D. Since $\angle C$ is given as $50^\circ$, and $\angle BCD$ is part of this configuration, the measure of $\angle BCD$ is $50^\circ$. 5. **Final answer:** $$\boxed{50^\circ}$$