Question

what is the measure of \\(\overarc{ced}\\)? 212° 148° 106° 108°

Explanation:

Step1: Recall circle angle sum

A full circle is \(360^\circ\), but we can also use the fact that the measure of an arc is the sum of the central angles subtended by its parts. Wait, actually, first, let's find the measure of arc \(CED\). The central angles for arcs: we have angle \( \angle CBE = 52^\circ \), angle \( \angle EBD = 160^\circ \)? Wait, no, wait. Wait, the center is \(B\). So arc \(CED\) is composed of arc \(CE\) and arc \(ED\)? Wait, no, maybe better: the total around a point is \(360^\circ\), but actually, the measure of arc \(CED\) can be found by adding the central angles that make it up. Wait, no, let's see: the central angle for arc \(CE\) is \(52^\circ\), and the central angle for arc \(ED\): wait, the angle \( \angle DBE \) – wait, the given angle is \(160^\circ\) between \(D\) and \(E\)? Wait, no, the diagram: \(B\) is the center. So \( \angle CBE = 52^\circ \), \( \angle EBD = 160^\circ \)? Wait, no, that can't be, because \(52 + 160 = 212\), but a straight line is \(180\). Wait, maybe I misread. Wait, the angle between \(B C\) and \(B E\) is \(52^\circ\), and between \(B E\) and \(B D\) is \(160^\circ\)? No, that's more than \(180\). Wait, no, maybe the angle between \(B D\) and \(B E\) is \(160^\circ\), and between \(B E\) and \(B C\) is \(52^\circ\), but then the angle between \(B C\) and \(B D\) would be \(52 + 160 = 212\), but that's more than \(180\), which is impossible. Wait, maybe the \(160^\circ\) is the angle between \(B D\) and \(B E\), and the angle between \(B C\) and \(B E\) is \(52^\circ\), so the angle between \(B C\) and \(B D\) is \(360 - 52 - 160 = 148^\circ\)? No, wait, no. Wait, the measure of an arc is equal to its central angle. So arc \(CED\): let's see, arc \(CE\) has central angle \(52^\circ\), arc \(ED\) has central angle... Wait, no, maybe the arc \(CED\) is the major arc? Wait, no, the question is about arc \(CED\). Wait, let's calculate the measure of arc \(CED\). The central angles: the angle for arc \(CE\) is \(52^\circ\), and the angle for arc \(ED\): wait, the straight line from \(D\) to \(E\) through \(B\)? No, \(B\) is the center. Wait, maybe the total around \(B\) is \(360^\circ\), but we can also use the fact that the measure of arc \(CED\) is the sum of arc \(CE\) and arc \(ED\). Wait, no, maybe I made a mistake. Wait, let's look at the options. The options are \(212^\circ\), \(148^\circ\), \(106^\circ\), \(108^\circ\). Wait, maybe the arc \(CED\) is a major arc, but no, wait, let's think again. Wait, the central angle for arc \(CD\) would be \(360 - 52 - 160 = 148^\circ\)? No, that doesn't make sense. Wait, no, the measure of arc \(CED\): let's see, arc \(CE\) is \(52^\circ\), arc \(ED\): the angle between \(E\) and \(D\) at \(B\) – wait, the given angle is \(160^\circ\) between \(D\) and \(E\)? Wait, maybe the angle \( \angle DBE = 160^\circ \), and \( \angle CBE = 52^\circ \), so the angle \( \angle CBD = 360 - 160 - 52 = 148^\circ \)? No, that's not right. Wait, no, the measure of an arc is equal to its central angle. So arc \(CED\): let's see, the arc from \(C\) to \(E\) to \(D\). So the central angles for \(CE\) is \(52^\circ\), and for \(ED\) is \(160^\circ\)? But \(52 + 160 = 212\), but that's one option. Wait, but that would be if \(E\) is between \(C\) and \(D\), but the diagram: \(C\), \(E\), \(D\) – wait, the center is \(B\). So \(B C\), \(B E\), \(B D\) are radii. So \( \angle CBE = 52^\circ \), \( \angle EBD = 160^\circ \). Then arc \(CED\) is arc \(CE\) (central angle \(52^\circ\)) plus arc \(ED\) (central angle \(160^\circ\))? But \(52 + 160 = 212\), which is one of the options. Wait, but that seems too big. Wait, no, maybe I got the arc wrong. Wait, maybe arc \(CED\) is the major arc, but no, the options include \(212\). Wait, let's check: a full circle is \(360\), so if we have arc \(CD\) as the minor arc, then arc \(CED\) would be \(360 - \) arc \(CD\). But arc \(CD\) would be \(360 - 52 - 160 = 148\), so arc \(CED\) would be \(360 - 148 = 212\)? Wait, no, that's not. Wait, no, the measure of arc \(CED\) is the sum of the central angles from \(C\) to \(E\) to \(D\). So \( \angle CBE = 52^\circ \) (arc \(CE\)) and \( \angle EBD = 160^\circ \) (arc \(ED\)), so arc \(CED = 52 + 160 = 212^\circ \). Wait, but that's option A. But let's confirm. The central angle for an arc is equal to the measure of the arc. So if \(B\) is the center, then arc \(CE\) has measure \(52^\circ\) (since \( \angle CBE = 52^\circ \)), and arc \(ED\) has measure \(160^\circ\) (since \( \angle EBD = 160^\circ \)). Therefore, arc \(CED\) is arc \(CE\) + arc \(ED = 52 + 160 = 212^\circ \).

Step2: Verify with full circle

Alternatively, the total around the center is \(360^\circ\). The arc opposite to \(CED\) (arc \(CD\)) would have central angle \(360 - 52 - 160 = 148^\circ\), but \(CED\) is the other arc, so \(52 + 160 = 212^\circ\), which matches the first option.

Answer:

\(212^\circ\) (the first option, 212°)