1. **Problem statement:** Given a circle with center O, lines BD and CE intersect at O. We know: - $\angle BOC = 116^\circ$ - $\angle BFC = 28^\circ$ We need to find the values of $\angle BOC$, $\angle DAC$, $\angle BFC$, $\angle ADE$, and $\angle AEC$. 2. **Known facts and formulas:** - $\angle BOC$ is a central angle. - $\angle BFC$ is an inscribed angle subtending the same arc as $\angle BOC$. - The measure of an inscribed angle is half the measure of the central angle subtending the same arc. - Opposite angles formed by two intersecting chords satisfy $\angle DAC + \angle BOC = 180^\circ$. - Angles subtended by the same chord are equal. 3. **Find $\angle BOC$:** Given directly as $116^\circ$. 4. **Find $\angle BFC$:** Since $\angle BFC$ is an inscribed angle subtending the same arc as $\angle BOC$, $$\angle BFC = \frac{1}{2} \times \angle BOC = \frac{1}{2} \times 116^\circ = 58^\circ.$$ 5. **Find $\angle DAC$:** $\angle DAC$ and $\angle BOC$ are vertical angles formed by intersecting chords BD and CE at O, so $$\angle DAC + \angle BOC = 180^\circ.$$ Therefore, $$\angle DAC = 180^\circ - 116^\circ = 64^\circ.$$ 6. **Find $\angle ADE$:** $\angle ADE$ subtends the same arc as $\angle AEC$, so they are equal. 7. **Find $\angle AEC$:** $\angle AEC$ is an inscribed angle subtending arc AC. Since $\angle ADE$ and $\angle AEC$ subtend the same arc, $$\angle ADE = \angle AEC.$$ **Final answers:** - $\angle BOC = 116^\circ$ - $\angle DAC = 64^\circ$ - $\angle BFC = 58^\circ$ - $\angle ADE = \angle AEC$ (equal angles, exact value depends on arc AC but are equal)