Question

what are the measures of angles 1 and 2?
m∠1 =
m∠2 = 47
50
53
100
(image of a circle with center u, points q, r, s, t on the circumference. arc qr is 53°, arc ts is 47°, lines qt and rs intersect inside the circle, forming angles 1 and 2.)

Explanation:

Step1: Find \( m\angle1 \)

The measure of an inscribed angle is half the measure of its intercepted arc. For \( \angle1 \), the intercepted arc is \( \overset{\frown}{RT} \)? Wait, no, looking at the diagram, \( \angle1 \) intercepts arc \( \overset{\frown}{QT} \)? Wait, actually, the inscribed angle theorem: the measure of an inscribed angle is half the sum of the measures of the intercepted arcs? Wait, no, if two chords intersect, the measure of the angle is half the sum of the intercepted arcs. Wait, in the diagram, chords \( QR \) and \( QS \)? Wait, no, points \( Q, S \) are on a chord, and \( R, T \) are on another chord, intersecting at the angle \( \angle1 \). Wait, the arcs given are \( 53^\circ \) (arc \( QR \)) and \( 47^\circ \) (arc \( ST \))? Wait, no, the inscribed angle theorem for an angle formed by two chords intersecting inside the circle: the measure of the angle is half the sum of the measures of the intercepted arcs. Wait, no, actually, when two chords intersect inside a circle, the measure of the angle is equal to half the sum of the measures of the intercepted arcs. Wait, but in this case, \( \angle1 \): let's see, the arcs intercepted by \( \angle1 \) would be arc \( RT \) and arc \( QS \)? Wait, maybe I made a mistake. Wait, the arc \( QR \) is \( 53^\circ \), arc \( ST \) is \( 47^\circ \). Wait, no, actually, the angle \( \angle1 \) is an inscribed angle? Wait, no, the center is \( U \), but \( \angle1 \) is formed by chords \( QR \) and \( QS \)? Wait, no, the points are \( Q, R, S, T \) on the circle. Chord \( QT \) and chord \( RS \) intersect at some point, forming \( \angle1 \). Wait, the arc \( QR \) is \( 53^\circ \), arc \( ST \) is \( 47^\circ \). Then, the measure of \( \angle1 \): when two chords intersect inside the circle, the angle is half the sum of the intercepted arcs. Wait, no, the formula is \( m\angle1=\frac{1}{2}(m\overset{\frown}{RT} + m\overset{\frown}{QS}) \)? Wait, maybe not. Wait, alternatively, if \( \angle1 \) is an inscribed angle intercepting arc \( \overset{\frown}{RT} \), but no, the arc \( QR \) is \( 53^\circ \), arc \( ST \) is \( 47^\circ \). Wait, maybe the total circumference is \( 360^\circ \), but no, let's think again. Wait, the angle \( \angle1 \): the inscribed angle theorem says that the measure of an inscribed angle is half the measure of its intercepted arc. Wait, if \( \angle1 \) intercepts arc \( \overset{\frown}{RT} \), but we have arc \( QR = 53^\circ \) and arc \( ST = 47^\circ \). Wait, maybe the arc \( RT \) is equal to arc \( QS \)? No, wait, maybe \( \angle1 \) is equal to the measure of the arc opposite? Wait, no, let's check the options. The options for \( m\angle2 \) are 47, 50, 53, 100. Wait, \( \angle2 \) is at the center? Wait, \( U \) is the center, so \( \angle2 \) is a central angle. The central angle is equal to the measure of its intercepted arc. Wait, if the arc \( QT \) is... Wait, maybe the sum of arcs \( QR \) (53°) and \( RT \) and \( ST \) (47°) and \( QS \) is 360°, but that's too complicated. Wait, alternatively, the measure of \( \angle1 \): since \( \angle1 \) is an inscribed angle intercepting arc \( \overset{\frown}{RT} \), but maybe the arc \( RT \) is 100°? No, wait, the options for \( m\angle2 \) include 100. Wait, \( \angle2 \) is a central angle, so its measure is equal to the measure of its intercepted arc. If \( \angle2 \) intercepts arc \( RT \), and the sum of arcs \( QR \) (53°) and \( ST \) (47°) is 100°, so arc \( RT \) would be 100°? Wait, no, 53 + 47 = 100? Wait, 53 + 47 = 100? Yes, 53 + 47 = 100. Then, \( \angle2 \) is a central angle, so \( m\angle2 = 100^\circ \). Then, \( \angle1 \): since \( \angle1 \) is an inscribed angle intercepting arc \( RT \) (100°)? No, wait, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. Wait, no, if two chords intersect inside the circle, the angle is half the sum of the intercepted arcs. Wait, \( \angle1 \) is formed by chords \( QR \) and \( QS \), intercepting arcs \( \overset{\frown}{RT} \) and \( \overset{\frown}{QS} \)? No, maybe I'm overcomplicating. Wait, the options for \( m\angle1 \): let's see, if \( \angle2 \) is 100°, then \( \angle1 \) would be half of 100? No, 50? Wait, the options for \( m\angle1 \) – wait, the user's diagram: the arc \( QR \) is 53°, arc \( ST \) is 47°. Then, the sum of these two arcs is 53 + 47 = 100. Then, the central angle \( \angle2 \) intercepts the arc that's the sum? Wait, no, maybe \( \angle2 \) is the central angle for the arc opposite to the sum. Wait, let's start over.

1. For \( \angle2 \): It's a central angle (since \( U \) is the center), so its measure equals the measure of its intercepted arc. The intercepted arc for \( \angle2 \) is the arc that's opposite to the sum of arcs \( QR \) (53°) and \( ST \) (47°), which is 53 + 47 = 100°. So \( m\angle2 = 100^\circ \).

2. For \( \angle1 \): It's an inscribed angle formed by two chords intersecting inside the circle. The measure of such an angle is half the sum of the measures of the intercepted arcs. Wait, no, the formula is \( m\angle1 = \frac{1}{2}(m\overset{\frown}{RT} + m\overset{\frown}{QS}) \)? No, wait, actually, when two chords intersect inside a circle, the measure of the angle is equal to half the sum of the measures of the intercepted arcs. Wait, but if \( \angle2 \) is 100°, which is the central angle for arc \( RT \), then arc \( RT = 100^\circ \). Then, the other arc intercepted by \( \angle1 \) – wait, maybe \( \angle1 \) intercepts arc \( RT \) (100°) and arc \( QS \)? No, maybe \( \angle1 \) is an inscribed angle intercepting arc \( RT \), but that would be half of 100, which is 50. Wait, the options for \( m\angle1 \) – wait, the user's dropdown for \( m\angle1 \) is being filled, and \( m\angle2 \) has options 47, 50, 53, 100. So if \( m\angle2 = 100^\circ \) (central angle), then \( m\angle1 \) is half of 100? No, 50? Wait, 53 + 47 = 100, so the central angle \( \angle2 \) is 100°, and the inscribed angle \( \angle1 \) is half of (53 + 47) = 50°? Wait, 53 + 47 = 100, half of that is 50. So \( m\angle1 = 50^\circ \), \( m\angle2 = 100^\circ \).

Step1: Determine \( m\angle2 \)

\( \angle2 \) is a central angle (since \( U \) is the center), so its measure equals the measure of its intercepted arc. The intercepted arc is the sum of arcs \( \overset{\frown}{QR} \) (53°) and \( \overset{\frown}{ST} \) (47°), so \( 53 + 47 = 100^\circ \). Thus, \( m\angle2 = 100^\circ \).

Step2: Determine \( m\angle1 \)

\( \angle1 \) is an angle formed by two chords intersecting inside the circle. The measure of such an angle is half the sum of the measures of the intercepted arcs (arcs \( \overset{\frown}{QR} \) and \( \overset{\frown}{ST} \) in this case). So \( m\angle1 = \frac{1}{2}(53 + 47) = \frac{1}{2}(100) = 50^\circ \).

Answer:

\( m\angle1 = 50^\circ \)
\( m\angle2 = 100^\circ \)