1. **Stating the problem:** We are given a circle with points B, C, and E on the circumference, and a triangle ADE outside the circle with angles 25° and 50° at point A. We need to find: a. The length of chord \(\overline{BC}\) b. The measure of angle \(\angle C\) c. The measure of angle \(\angle CEB\) 2. **Analyzing the given information:** - The angle at B is given as 100°. - Triangle ADE has angles 25° and 50° at A, so the third angle at D is \(180^\circ - 25^\circ - 50^\circ = 105^\circ\). - Points B, C, E lie on the circle, so angles subtended by the same chord are related. 3. **Finding \(\overline{BC}\):** Since no lengths are given, we cannot find the exact length of \(\overline{BC}\) without additional data. The problem likely expects an angle or relationship rather than numeric length. 4. **Finding \(\angle C\):** - \(\angle B = 100^\circ\) is given. - In triangle BDC inscribed in the circle, the sum of angles is 180°. - If \(\angle B = 100^\circ\), then \(\angle C + \angle D = 80^\circ\). 5. **Finding \(\angle CEB\):** - Since B, C, E lie on the circle, \(\angle CEB\) is an inscribed angle subtending arc CB. - The measure of an inscribed angle is half the measure of its intercepted arc. - If \(\angle B = 100^\circ\), then arc BC is 200° (since central angle is twice inscribed angle). - Therefore, \(\angle CEB = \frac{1}{2} \times 200^\circ = 100^\circ\). **Final answers:** - a. Cannot determine \(\overline{BC}\) length without more data. - b. \(\angle C = 80^\circ - \angle D\) (depends on \(\angle D\)) - c. \(\angle CEB = 100^\circ\)