Question

in the circle below, \\( \overline{qs} \\) is a diameter. suppose \\( m\overarc{qr} = 46^\circ \\) and \\( m\angle qrt = 58^\circ \\). find the following.\\(\text{(a) } m\angle rqs = \square^\circ\\)\\(\text{(b) } m\angle srt = \square^\circ\\)

Explanation:

Step1: Find \( m\angle RQS \)

In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. For \( \angle RQS \), it intercepts arc \( \overset{\frown}{RS} \)? Wait, no, \( \angle RQS \) intercepts arc \( \overset{\frown}{RS} \)? Wait, actually, \( \angle RQS \) is an inscribed angle intercepting arc \( \overset{\frown}{RS} \)? Wait, no, the arc \( \overset{\frown}{QR} = 46^\circ \). Wait, \( \triangle QRS \): since \( QS \) is a diameter, \( \angle QRS = 90^\circ \) (angle inscribed in a semicircle). Wait, no, let's correct. The inscribed angle theorem: the measure of an inscribed angle is half the measure of its intercepted arc. So \( \angle RQS \) intercepts arc \( \overset{\frown}{RS} \)? Wait, no, \( \angle RQS \) is formed by chords \( RQ \) and \( SQ \), so it intercepts arc \( \overset{\frown}{RS} \). Wait, but we know arc \( \overset{\frown}{QR} = 46^\circ \). Wait, maybe another approach. In \( \triangle QRT \), we know \( \angle QRT = 58^\circ \), but maybe first part (a): \( \angle RQS \) is an inscribed angle intercepting arc \( \overset{\frown}{RS} \)? Wait, no, the arc \( \overset{\frown}{QR} = 46^\circ \), so the central angle for \( \overset{\frown}{QR} \) is \( 46^\circ \), so the inscribed angle \( \angle RQS \): wait, no, \( \angle RQS \) is an inscribed angle intercepting arc \( \overset{\frown}{RS} \)? Wait, I think I made a mistake. Let's recall: the measure of an inscribed angle is half the measure of its intercepted arc. So for \( \angle RQS \), the intercepted arc is \( \overset{\frown}{RS} \)? No, \( \angle RQS \) is at point \( Q \), between \( RQ \) and \( SQ \), so the intercepted arc is \( \overset{\frown}{RS} \). But \( QS \) is a diameter, so the arc \( \overset{\frown}{QS} \) is \( 180^\circ \). So arc \( \overset{\frown}{QR} + \overset{\frown}{RS} = 180^\circ \)? Wait, no, \( \overset{\frown}{QR} + \overset{\frown}{RS} = \overset{\frown}{QS} \), which is a semicircle, so \( 180^\circ \). Wait, but we know \( m\overset{\frown}{QR} = 46^\circ \), so \( m\overset{\frown}{RS} = 180^\circ - 46^\circ = 134^\circ \)? No, that can't be. Wait, no, \( QS \) is a diameter, so the arc \( \overset{\frown}{QS} \) is \( 180^\circ \), so arc \( \overset{\frown}{QR} + \overset{\frown}{RS} = 180^\circ \). So \( \overset{\frown}{RS} = 180^\circ - 46^\circ = 134^\circ \). Then the inscribed angle \( \angle RQS \) intercepts arc \( \overset{\frown}{RS} \)? No, that would be \( \frac{1}{2} \times 134^\circ = 67^\circ \)? Wait, no, that doesn't seem right. Wait, maybe the first part (a) is using the inscribed angle theorem correctly. Wait, the inscribed angle \( \angle RQS \) intercepts arc \( \overset{\frown}{RS} \)? No, \( \angle RQS \) is formed by chords \( RQ \) and \( SQ \), so the intercepted arc is \( \overset{\frown}{RS} \). Wait, but maybe I mixed up. Let's try part (a) again. The arc \( \overset{\frown}{QR} = 46^\circ \), so the inscribed angle \( \angle RQS \) is half of the arc it intercepts? Wait, no, \( \angle RQS \) is an inscribed angle intercepting arc \( \overset{\frown}{RS} \), but we need to find another way. Wait, maybe the triangle \( QRT \): \( \angle QRT = 58^\circ \), but maybe that's for part (b). Wait, no, let's do part (a) first. In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. So \( \angle RQS \) intercepts arc \( \overset{\frown}{RS} \), but we know arc \( \overset{\frown}{QR} = 46^\circ \), and \( QS \) is a diameter, so arc \( \overset{\frown}{QS} = 180^\circ \), so arc \( \overset{\frown}{RS} = 180^\circ - 46^\circ = 134^\circ \). Then \( \angle RQS = \frac{1}{2} \times \) arc \( \overset{\frown}{RS} \)? No, that would be \( 67^\circ \), but that doesn't seem right. Wait, maybe I got the intercepted arc wrong. \( \angle RQS \) is formed by chords \( RQ \) and \( SQ \), so the intercepted arc is \( \overset{\frown}{RS} \), but actually, no: the inscribed angle's measure is half the measure of its intercepted arc. So if \( \angle RQS \) is at point \( Q \), between \( RQ \) and \( SQ \), then the intercepted arc is \( \overset{\frown}{RS} \). But maybe another approach: in \( \triangle QRS \), since \( QS \) is a diameter, \( \angle QRS = 90^\circ \) (angle inscribed in a semicircle). Wait, yes! That's the key. So \( \angle QRS = 90^\circ \), because any triangle inscribed in a semicircle is a right triangle, with the right angle at the point on the circle. So \( \angle QRS = 90^\circ \). Then in \( \triangle QRS \), we have angles: \( \angle QRS = 90^\circ \), \( \angle RQS \), and \( \angle RSQ \). But we also know arc \( \overset{\frown}{QR} = 46^\circ \), so the inscribed angle \( \angle RSQ \) intercepts arc \( \overset{\frown}{QR} \), so \( \angle RSQ = \frac{1}{2} \times 46^\circ = 23^\circ \). Then in \( \triangle QRS \), \( \angle RQS + \angle RSQ + \angle QRS = 180^\circ \), so \( \angle RQS + 23^\circ + 90^\circ = 180^\circ \), so \( \angle RQS = 180^\circ - 90^\circ - 23^\circ = 67^\circ \)? Wait, that can't be, because 23 + 90 is 113, 180 - 113 is 67. But let's check again. The inscribed angle \( \angle RSQ \) intercepts arc \( \overset{\frown}{QR} \), so \( m\angle RSQ = \frac{1}{2} m\overset{\frown}{QR} = \frac{1}{2} \times 46^\circ = 23^\circ \). Then in \( \triangle QRS \), \( \angle QRS = 90^\circ \) (since \( QS \) is a diameter, angle in semicircle), so \( \angle RQS = 180^\circ - 90^\circ - 23^\circ = 67^\circ \). Wait, but that seems high. Wait, maybe I made a mistake here. Let's try another way. The arc \( \overset{\frown}{QR} = 46^\circ \), so the central angle for \( \overset{\frown}{QR} \) is \( 46^\circ \), so the inscribed angle \( \angle RQS \) is half of that? No, \( \angle RQS \) is not intercepting \( \overset{\frown}{QR} \), it's intercepting \( \overset{\frown}{RS} \). Wait, maybe the problem is simpler. Let's look at part (a): \( \angle RQS \) is an inscribed angle intercepting arc \( \overset{\frown}{QR} \)? No, \( \angle RQS \) is formed by chords \( RQ \) and \( SQ \), so the intercepted arc is \( \overset{\frown}{RS} \). Wait, I think I confused the intercepted arc. Let's recall: the inscribed angle theorem states that an angle \( \theta \) formed by two chords in a circle with the vertex on the circle is equal to half the sum or half the difference of the intercepted arcs, depending on the position. Wait, no, if the angle is an inscribed angle (vertex on the circle, sides as chords), then it's half the measure of its intercepted arc. So \( \angle RQS \) has vertex at \( Q \), sides \( RQ \) and \( SQ \), so it intercepts arc \( \overset{\frown}{RS} \). So \( m\angle RQS = \frac{1}{2} m\overset{\frown}{RS} \). But we know \( QS \) is a diameter, so \( m\overset{\frown}{QS} = 180^\circ \), and \( m\overset{\frown}{QR} = 46^\circ \), so \( m\overset{\frown}{RS} = m\overset{\frown}{QS} - m\overset{\frown}{QR} = 180^\circ - 46^\circ = 134^\circ \). Then \( m\angle RQS = \frac{1}{2} \times 134^\circ = 67^\circ \). Okay, that matches the previous result. So part (a) is 67? Wait, but let's check part (b). For \( m\angle SRT \), we know \( \angle QRT = 58^\circ \), and \( \angle QRS = 90^\circ \) (from part (a), since \( QS \) is a diameter). So \( \angle SRT = \angle QRS - \angle QRT = 90^\circ - 58^\circ = 32^\circ \). Wait, is that correct? Let's see: \( \angle QRS = 90^\circ \) (angle in semicircle), \( \angle QRT = 58^\circ \), so \( \angle SRT = 90^\circ - 58^\circ = 32^\circ \). Yes, that makes sense.

Step1 (a): Calculate \( m\angle RQS \)

The arc \( \overset{\frown}{QR} = 46^\circ \), and \( QS \) is a diameter, so \( m\overset{\frown}{QS} = 180^\circ \). Thus, \( m\overset{\frown}{RS} = 180^\circ - 46^\circ = 134^\circ \). By the inscribed angle theorem, \( m\angle RQS = \frac{1}{2} m\overset{\frown}{RS} = \frac{1}{2} \times 134^\circ = 67^\circ \).

Step2 (b): Calculate \( m\angle SRT \)

Since \( QS \) is a diameter, \( \angle QRS = 90^\circ \) (angle inscribed in a semicircle). We know \( m\angle QRT = 58^\circ \), so \( m\angle SRT = m\angle QRS - m\angle QRT = 90^\circ - 58^\circ = 32^\circ \).

Answer:

(a) \( \boldsymbol{67} \)
(b) \( \boldsymbol{32} \)