Question

in circle o, \\(\overline{ac}\\) and \\(\overline{be}\\) are diameters. the measure of arc dc is \\(50^\circ\\). what is the measure of \\(\overline{ebc}\\)? \\(\bigcirc\\ 40^\circ\\) \\(\bigcirc\\ 90^\circ\\) \\(\bigcirc\\ 140^\circ\\) \\(\bigcirc\\ 220^\circ\\)

Explanation:

Step1: Find arc AB measure

Since $\overline{AC}$ is a diameter, $\text{arc } ABC = 180^\circ$. Given $\text{arc } DC = 50^\circ$ and $\angle DOC = 90^\circ$ (right angle), $\text{arc } AB = \text{arc } DC = 50^\circ$ (vertical angles correspond to equal arcs).

Step2: Calculate arc EB measure

$\overline{BE}$ is a diameter, so $\text{arc } BE = 180^\circ$. $\text{arc } EB = \text{arc } EA + \text{arc } AB$, so $\text{arc } EA = 180^\circ - 50^\circ = 130^\circ$.

Step3: Find arc EBC measure

$\text{arc } EBC = \text{arc } EB + \text{arc } BC$. $\text{arc } BC = 180^\circ - 50^\circ = 130^\circ$? No, correct: $\text{arc } EBC = \text{arc } EA + \text{arc } AC$. $\text{arc } AC = 180^\circ$, $\text{arc } EA = 40^\circ$ (correction: $\angle AOE = \angle BOC$, $\angle BOC = 90^\circ - 50^\circ = 40^\circ$, so $\text{arc } EA = 40^\circ$). Then $\text{arc } EBC = \text{arc } EA + \text{arc } AC = 40^\circ + 180^\circ = 220^\circ$.

Answer:

D. $220^\circ$