Question

question 3 of 44
what is the measure of ∠dbe?
image of a circle with center b, points d, e, a, c; arc de labeled 45°, arc ac labeled 75°; text click here for long description
options:
○ a. 60°
○ b. 75°
○ c. 45°
○ d. 120°

Explanation:

Step1: Identify the central angles

The sum of all central angles in a circle is \(360^\circ\), but we can also use the fact that vertical angles are equal and adjacent central angles on a straight line sum to \(180^\circ\). Wait, actually, looking at the arcs: arc \(DE\) is \(45^\circ\), arc \(AC\) is \(75^\circ\)? Wait, no, maybe \(B\) is the center. So \(\angle DBE\) is a central angle, but wait, no—wait, the arcs: arc \(DE\) is \(45^\circ\), but maybe we need to find the measure of \(\angle DBE\) as a central angle? Wait, no, maybe the total around point \(B\) (the center) should sum to \(360^\circ\), but actually, the key is that the sum of arcs \(DE\), \(EA\), \(AC\), and \(CD\) is \(360^\circ\), but maybe we can use the fact that opposite arcs: arc \(AC\) is \(75^\circ\), so its vertical angle arc would be equal, but wait, no. Wait, maybe the measure of \(\angle DBE\) is related to the arcs. Wait, no, actually, the central angle \(\angle DBE\) is equal to the measure of arc \(DE\)? No, that can't be. Wait, maybe I made a mistake. Wait, let's think again. The circle has center \(B\), so \(BD\), \(BE\), \(BA\), \(BC\) are radii. The arc \(DE\) is \(45^\circ\), arc \(AC\) is \(75^\circ\). Then, the sum of arcs \(DE\), \(EA\), \(AC\), and \(CD\) is \(360^\circ\), but also, the vertical angles: \(\angle DBE\) and \(\angle ABC\) are vertical angles? No, wait, \(D\) and \(E\) are on the circle, \(A\) and \(C\) are on the circle. So the central angle \(\angle DBE\) is formed by radii \(BD\) and \(BE\), so its measure is equal to the measure of arc \(DE\)? But arc \(DE\) is \(45^\circ\)? No, that's not matching the options. Wait, maybe the total of the arcs on one side: the sum of arc \(DE\) (45°) and arc \(AC\) (75°) and the other two arcs? Wait, no, maybe the straight line: the sum of angles around a point is \(360^\circ\), but the sum of two adjacent central angles on a straight line is \(180^\circ\). Wait, maybe the measure of \(\angle DBE\) is \(180^\circ - 45^\circ - 75^\circ\)? Let's calculate that: \(180 - 45 - 75 = 60^\circ\). Ah, that makes sense. Because the straight line (diameter) would have a central angle of \(180^\circ\), so the sum of arc \(DE\) (45°), arc \(AC\) (75°), and the angle \(\angle DBE\) (wait, no, \(\angle DBE\) is a central angle, and the other arcs: wait, maybe the two arcs \(DE\) (45°) and \(AC\) (75°) are on one side, and the other two arcs are equal to \(\angle DBE\) and its vertical angle? Wait, no, let's do it step by step.

Step1: Recall that the sum of angles on a straight line is \(180^\circ\)

The line \(AC\) and \(DE\) intersect at the center \(B\), so the sum of the central angles around a straight line (e.g., along \(AC\) or \(DE\)) is \(180^\circ\). Wait, actually, the measure of \(\angle DBE\) can be found by subtracting the measures of the other two arcs (or central angles) from \(180^\circ\). The arc \(DE\) is \(45^\circ\), the arc \(AC\) is \(75^\circ\), so the remaining angle ( \(\angle DBE\)) is \(180^\circ - 45^\circ - 75^\circ\).

Step2: Calculate the measure

\(180 - 45 - 75 = 60\) degrees.

Answer:

A. 60°