Question

what is the measure of angle aoc? 66° 42° 58°

Explanation:

Step1: Analyze triangle OBC

First, we know that angle at B (∠DBE or ∠OBC's vertical angle) is 58°, but wait, actually, let's look at the straight line DC. Wait, maybe we can use the fact that in triangle OBC, or maybe vertical angles? Wait, no, let's check the arcs. Wait, the arc DE is 50°, angle at B is 58°. Wait, maybe we can find angle OCB or something. Wait, alternatively, let's consider that angle AOC: let's find the measure. Wait, maybe we can use the fact that the sum of angles or arcs. Wait, another approach: in the diagram, we have a circle with center O? Wait, no, B is a point on DC, maybe DC is a chord, and O is the center? Wait, OA, OC are radii? Wait, maybe the key is to find angle AOC. Let's see, angle at B: ∠DBE is 58°, so ∠OBC is also 58° (vertical angles). Then, in triangle OBC, if we can find other angles. Wait, maybe OC and OB are radii? Wait, no, maybe not. Wait, maybe the arc DE is 50°, so the central angle for DE would be 50°, but angle at B is 58°. Wait, maybe we can calculate angle AOC as follows: Let's assume that DC is a straight line, so angle DBE is 58°, so angle OBC is 58°. Then, maybe triangle OBC has some properties. Wait, alternatively, let's look at the options. The options are 66°, 42°, 58°. Wait, maybe we can use the fact that angle AOC is equal to 180° - 58° - 56°? No, that doesn't make sense. Wait, maybe another way: the arc from A to C? Wait, maybe the measure of angle AOC is 66°. Wait, let's re-examine. Wait, the angle at B is 58°, arc DE is 50°, so maybe the central angle for arc AE? Wait, no. Wait, maybe the correct approach is: in the diagram, we have angle at B is 58°, and arc DE is 50°, so the inscribed angle or central angle. Wait, maybe the answer is 66°, but let's check. Wait, maybe the sum of angles: 180 - 58 - 56? No. Wait, maybe the correct answer is 66°, but let's think again. Wait, perhaps the key is that angle AOC is calculated as 180° - 58° - 56°? No, maybe not. Wait, maybe the answer is 66°, so let's go with that. Wait, no, maybe I made a mistake. Wait, let's try again. Let's assume that O is the center, so OA, OC, OD, OE are radii. Then, arc DE is 50°, so central angle DOE is 50°. Then, angle DBE is 58°, which is an inscribed angle? No, DBE is at point B, not on the circle. Wait, maybe B is the intersection of AE and DC. So AE and DC intersect at B, with angle DBE = 58°, arc DE = 50°. Then, the measure of angle AOC can be found by the formula: angle AOC = 180° - angle DBE - arc DE / 2? No, that's not right. Wait, the formula for the angle formed by two chords intersecting at a point is equal to half the sum of the intercepted arcs. So angle DBE is formed by chords DE and AC? Wait, no, chords DE and AE? Wait, angle at B: ∠DBE is formed by chords DB and EB, intercepting arc DE and arc AC? Wait, the measure of an angle formed by two chords intersecting at a point inside the circle is equal to half the sum of the measures of the intercepted arcs. So ∠DBE = 1/2 (arc DE + arc AC). We know ∠DBE = 58°, arc DE = 50°, so 58° = 1/2 (50° + arc AC). Solving for arc AC: 58° * 2 = 50° + arc AC → 116° = 50° + arc AC → arc AC = 66°. Then, the central angle for arc AC is angle AOC, which is equal to the measure of arc AC, so angle AOC = 66°. Ah, that makes sense. So using the intersecting chords angle theorem: the measure of an angle formed by two chords intersecting inside the circle is equal to half the sum of the measures of the intercepted arcs. So ∠DBE = 1/2 (arc DE + arc AC). We have ∠DBE = 58°, arc DE = 50°, so solving for arc AC gives 66°, so the central angle AOC (which intercepts arc AC) is 66°.

Step2: Confirm the calculation

Using the intersecting chords angle theorem:
\( \angle DBE = \frac{1}{2} (\text{arc } DE + \text{arc } AC) \)
Given \( \angle DBE = 58^\circ \) and \( \text{arc } DE = 50^\circ \), substitute:
\( 58^\circ = \frac{1}{2} (50^\circ + \text{arc } AC) \)
Multiply both sides by 2:
\( 116^\circ = 50^\circ + \text{arc } AC \)
Subtract \( 50^\circ \) from both sides:
\( \text{arc } AC = 66^\circ \)
Since \( \angle AOC \) is the central angle intercepting arc \( AC \), \( \angle AOC = \text{arc } AC = 66^\circ \).

Answer:

\( 66^\circ \)