Question

in circle z, what is m∠2? 70° 133° 140° 147°

Explanation:

Step1: Find the measure of the arc opposite to the known arcs

The total measure of a circle is \( 360^\circ \). We know two arcs: \( 133^\circ \) and \( 147^\circ \). Let the measure of the remaining arc (arc \( AB \) or arc \( CD \) related to the angle) be \( x \). So, \( 133 + 147 + x + \text{the other arc} = 360 \). Wait, actually, when two chords intersect, the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs. But first, let's find the measure of the arc that is supplementary to the arcs we have. Wait, the sum of the measures of the arcs around a circle is \( 360^\circ \). So, the sum of the two given arcs is \( 133^\circ + 147^\circ = 280^\circ \). Then the sum of the other two arcs is \( 360^\circ - 280^\circ = 80^\circ \)? Wait, no, maybe I made a mistake. Wait, actually, when two chords intersect inside a circle, the measure of the angle is half the sum of the measures of the intercepted arcs. But first, let's find the measure of the arc that is opposite to the angle \( \angle 2 \). Wait, the total circumference is \( 360^\circ \). So, the sum of the arcs \( 133^\circ \) and \( 147^\circ \) is \( 133 + 147 = 280^\circ \). Then the remaining two arcs (let's say arc \( AC \) and arc \( BD \)) sum to \( 360 - 280 = 80^\circ \)? Wait, no, maybe the angle \( \angle 2 \) is formed by two intersecting chords, so the measure of \( \angle 2 \) is half the sum of the measures of the arcs intercepted by \( \angle 2 \) and its vertical angle. Wait, no, actually, when two chords intersect inside a circle, the measure of an angle formed is equal to half the sum of the measures of the intercepted arcs. Wait, but maybe the arcs given are \( 133^\circ \) and \( 147^\circ \), and the other two arcs: let's calculate the measure of the arc that is not \( 133^\circ \) or \( 147^\circ \). Wait, the total is \( 360^\circ \), so \( 360 - 133 - 147 = 80^\circ \)? Wait, no, that can't be. Wait, maybe the arcs are \( 133^\circ \), \( 147^\circ \), and two other arcs. Wait, perhaps I misread the diagram. Wait, the diagram shows a circle with center Z, and two chords intersecting, forming angles \( \angle 1 \) and \( \angle 2 \). The arcs given are \( 133^\circ \) (arc \( AB \)) and \( 147^\circ \) (arc \( CD \)). Wait, when two chords intersect inside a circle, the measure of the angle is half the sum of the measures of the intercepted arcs. So, for angle \( \angle 2 \), the intercepted arcs are the arc opposite to it and the arc adjacent? Wait, no, the formula is: if two chords intersect at a point inside the circle, then the measure of the angle is half the sum of the measures of the intercepted arcs. So, let's denote the arcs: let arc \( AB = 133^\circ \), arc \( CD = 147^\circ \), and the other two arcs (arc \( AC \) and arc \( BD \)) be \( x \) and \( y \). Then \( 133 + 147 + x + y = 360 \), so \( x + y = 360 - 133 - 147 = 80^\circ \). Now, angle \( \angle 2 \) is formed by chords \( AC \) and \( BD \), so the measure of \( \angle 2 \) is half the sum of the measures of arc \( AB \) and arc \( CD \)? Wait, no, that's not right. Wait, actually, when two chords intersect inside a circle, the measure of the angle is half the sum of the measures of the intercepted arcs. So, if the angle is \( \angle 2 \), the intercepted arcs are the arcs that are cut off by the two chords and lie on either side of the angle. Wait, maybe I got the formula wrong. Let me recall: the measure of an angle formed by two intersecting chords is equal to half the sum of the measures of the intercepted arcs. So, if the two chords are \( AB \) and \( CD \) intersecting at point \( E \) (in the diagram, the intersection point is where \( \angle 1 \) and \( \angle 2 \) are), then \( \angle 1 \) and \( \angle 2 \) are vertical angles? No, \( \angle 1 \) and \( \angle 2 \) are adjacent angles, supplementary? Wait, no, in the diagram, \( \angle 1 \) and \( \angle 2 \) are adjacent angles formed by two intersecting chords, so they are supplementary? Wait, no, if two lines intersect, the adjacent angles are supplementary. Wait, but in a circle, when two chords intersect, the vertical angles are equal, and each pair of adjacent angles are supplementary. Wait, but the arcs: the measure of \( \angle 1 \) is half the sum of the measures of the arcs intercepted by \( \angle 1 \) and its vertical angle. Wait, maybe the given arcs are \( 133^\circ \) and \( 147^\circ \), and the angle \( \angle 2 \) is formed such that the sum of the arcs intercepted by \( \angle 2 \) and its vertical angle is \( 133 + 147 = 280^\circ \), so the measure of \( \angle 2 \) is half of that? No, that would be \( 140^\circ \), because \( (133 + 147)/2 = 280/2 = 140^\circ \). Ah, that makes sense. So, the formula for the measure of an angle formed by two intersecting chords inside a circle is \( \frac{1}{2} \times (\text{measure of arc 1} + \text{measure of arc 2}) \), where arc 1 and arc 2 are the arcs intercepted by the angle and its vertical angle. So, in this case, the two arcs intercepted by \( \angle 2 \) and its vertical angle are \( 133^\circ \) and \( 147^\circ \). So, \( m\angle 2 = \frac{1}{2} \times (133 + 147) \).

Step2: Calculate the measure of \( \angle 2 \)

First, sum the measures of the two arcs: \( 133^\circ + 147^\circ = 280^\circ \). Then, take half of that sum: \( \frac{280^\circ}{2} = 140^\circ \).

Answer:

\( 140^\circ \) (corresponding to the option "140°")