Question

what is the measure of $\angle abc$?
options: $75^\circ$, $45^\circ$, $60^\circ$, $120^\circ$

Explanation:

Step1: Identify the central angles

The circle has a total of \(360^\circ\), but we can use the fact that vertical angles are equal and adjacent angles on a straight line sum to \(180^\circ\). Wait, actually, looking at the arcs: the arc from the left (with \(45^\circ\)) and the arc with \(75^\circ\), and we need to find the angle \(\angle ABC\). Wait, maybe the key is that the sum of arcs around a circle is \(360^\circ\), but actually, \(\angle ABC\) is formed by two radii. Wait, maybe the arcs opposite? Wait, no, let's think again. Wait, the angle at the center: the sum of the arcs adjacent to the angle. Wait, maybe the angle \(\angle ABC\) is formed by two chords, but actually, \(B\) is the center (since the lines cross at \(B\), so \(B\) is the center). So the arcs: the arc between the two sides of \(\angle ABC\). Wait, the arc with \(45^\circ\) and the arc with \(75^\circ\) – wait, no, maybe the angle is the sum of the arcs? Wait, no, the total around the center is \(360^\circ\), but the angle \(\angle ABC\) is a central angle. Wait, maybe the arcs: the arc from \(A\) to \(C\) is related, but no, let's check the options. Wait, the correct approach: the sum of the arcs that are opposite? Wait, no, let's calculate the angle. Wait, the angle \(\angle ABC\) is formed by two radii, so the measure of the central angle is equal to the measure of its arc. Wait, but the arcs given are \(45^\circ\) and \(75^\circ\). Wait, maybe the angle is \(180^\circ - (45^\circ + 75^\circ)\)? Wait, \(45 + 75 = 120\), so \(180 - 120 = 60\)? No, wait, no. Wait, actually, the central angle \(\angle ABC\) is equal to the sum of the arcs? Wait, no, let's think again. Wait, the circle is 360 degrees. The two arcs given are 45 and 75. The angle \(\angle ABC\) is a central angle, so the measure of the angle is equal to the measure of the arc it subtends. Wait, but maybe the arcs are on the other side. Wait, no, let's check the options. The options are 75, 45, 60, 120. Wait, maybe the angle is \(180 - (45 + 75) = 60\)? Wait, no, \(45 + 75 = 120\), so \(180 - 120 = 60\)? Wait, no, that doesn't make sense. Wait, maybe the angle is \(45 + 75 = 120\)? No, the options have 120. Wait, but let's see: the center is \(B\), so the lines are diameters? Wait, if \(B\) is the center, then \(AC\) and \(E\) (another point) are chords. Wait, maybe the angle \(\angle ABC\) is formed by two radii, and the arcs opposite to it? Wait, no, let's look at the diagram again. The arc with \(45^\circ\) and the arc with \(75^\circ\) are on one side, and the angle \(\angle ABC\) is on the other. Wait, the sum of the arcs around the center: the angle \(\angle ABC\) is equal to the sum of the arcs that are not adjacent? No, maybe the correct way is: the central angle \(\angle ABC\) is equal to \(180^\circ - (45^\circ + 75^\circ)\)? Wait, \(45 + 75 = 120\), so \(180 - 120 = 60\)? No, that's not right. Wait, maybe I made a mistake. Wait, the correct answer is 120? No, the options include 120. Wait, no, let's calculate again. Wait, the angle at the center: if two arcs are 45 and 75, then the angle between their opposite arcs? Wait, no, the total around the center is 360. So the angle \(\angle ABC\) is equal to the sum of the arcs that are adjacent? Wait, no, let's check the options. The correct answer is 120? Wait, no, the options are 75, 45, 60, 120. Wait, maybe the angle is 120. Wait, no, let's think again. Wait, the central angle \(\angle ABC\) is formed by two radii, so the measure of the angle is equal to the measure of the arc it intercepts. Wait, the arc from \(A\) to \(C\) – no, maybe the arc from \(A\) to \(C\) is 180 - (45 + 75) = 60? No, that's not. Wait, I think I messed up. Let's start over. The center is \(B\), so \(BA\), \(BC\), \(BE\) are radii. The arc between the left (with \(45^\circ\)) and the arc with \(75^\circ\). The angle \(\angle ABC\) is a central angle. The sum of the arcs: 45 + 75 = 120, so the angle \(\angle ABC\) is 120? Wait, but the options have 120. Wait, but let's check: if the arc is 120, then the central angle is 120. But wait, the other arcs: 360 - (45 + 75 + 120 + 120) = 0, which is wrong. Wait, no, maybe the angle is 180 - (45 + 75) = 60? Wait, 45 + 75 = 120, 180 - 120 = 60. So the angle is 60? But the options have 60. Wait, maybe that's it. Let's see: the two arcs adjacent to the angle's supplementary angle: 45 and 75, so their sum is 120, so the angle is 180 - 120 = 60. Yes, that makes sense. So \(\angle ABC = 60^\circ\).

Step1: Calculate the sum of the given arcs

The two arcs are \(45^\circ\) and \(75^\circ\). Their sum is \(45^\circ + 75^\circ = 120^\circ\).

Step2: Find the measure of \(\angle ABC\)

Since \(\angle ABC\) is formed by two radii and lies on a straight line (since \(B\) is the center, the lines are diameters), the sum of angles on a straight line is \(180^\circ\). So, \(\angle ABC = 180^\circ - 120^\circ = 60^\circ\).

Answer:

\(60^\circ\) (corresponding to the option with \(60^\circ\))