Question

what is the measure of \\(\overarc{sq}\\)?

75°
s
r
95°
7
q

○ 95
○ 190
○ 265
○ 360

Explanation:

Step1: Recall Inscribed Quadrilateral Property

In a cyclic quadrilateral (quadrilateral inscribed in a circle), the measure of an inscribed angle is half the measure of its intercepted arc. Also, the sum of opposite angles in a cyclic quadrilateral is \(180^\circ\), but here we use the property that the measure of an arc intercepted by an inscribed angle is twice the angle. Wait, actually, for a chord, the inscribed angle over it: but here, the angle at \(S\) between \(SR\) and \(SQ\) is \(95^\circ\), but wait, no—wait, the arc \(SR\) is \(75^\circ\), and we need to find arc \(SQ\). Wait, no, maybe the quadrilateral is cyclic, so the angle at \(S\) is an inscribed angle? Wait, no, the angle given is \(95^\circ\) at \(S\) between \(SQ\) and \(SR\). Wait, actually, in a circle, the measure of an inscribed angle is half the measure of its intercepted arc. But if we consider the angle at \(S\) as an inscribed angle, but no—wait, maybe the arc \(SQ\) is related to the angle. Wait, no, the key is that in a circle, the measure of a central angle is equal to its arc, and inscribed angle is half. But here, maybe the angle at \(S\) is an inscribed angle intercepting arc \(QR\), but no, the problem is about arc \(SQ\). Wait, maybe I made a mistake. Wait, the length of \(SQ\) is 7? No, the 7 is the length? Wait, no, the diagram: \(S\), \(R\), \(Q\) on the circle, with arc \(SR = 75^\circ\), angle at \(S\) is \(95^\circ\). Wait, actually, in a cyclic quadrilateral, the sum of the measures of opposite arcs? No, wait, the measure of an inscribed angle is half the measure of its intercepted arc. So if angle at \(S\) is \(95^\circ\), and it's an inscribed angle intercepting arc \(QR\), but we need arc \(SQ\). Wait, no, maybe the angle at \(S\) is formed by two chords \(SQ\) and \(SR\), so the angle between them is an inscribed angle, and the intercepted arc is \(QR\). But we know arc \(SR = 75^\circ\), and the total circumference is \(360^\circ\). Wait, no, maybe the angle at \(S\) is \(95^\circ\), so the arc \(SQ\) is twice the angle? No, wait, no—wait, the formula for the measure of an arc intercepted by an inscribed angle: \( \text{arc measure} = 2 \times \text{inscribed angle} \). Wait, if the angle at \(S\) is \(95^\circ\), and it's an inscribed angle intercepting arc \(QR\), then arc \(QR = 2 \times 95^\circ = 190^\circ\)? No, that can't be. Wait, no, maybe the angle at \(S\) is a central angle? No, central angle is at the center. Wait, maybe the diagram is a cyclic quadrilateral, so the sum of arc \(SQ\) and arc \(SR\) and the other arcs? Wait, no, let's think again. Wait, the problem is asking for the measure of arc \(SQ\). The options are 95, 190, 265, 360. Let's recall that in a circle, the measure of an inscribed angle is half the measure of its intercepted arc. So if the angle at \(S\) is \(95^\circ\), and it's an inscribed angle, then the arc it intercepts (opposite arc) would be \(2 \times 95^\circ = 190^\circ\). Wait, that makes sense. So arc \(SQ\) is \(190^\circ\)? Wait, no, wait: if the angle at \(S\) is \(95^\circ\), and it's an inscribed angle intercepting arc \(QR\), then arc \(QR = 190^\circ\), but we need arc \(SQ\). Wait, no, maybe the angle at \(S\) is between \(SQ\) and \(SR\), so the arc \(SQ\) is related to the angle. Wait, maybe I messed up. Wait, the arc \(SR\) is \(75^\circ\), and the angle at \(S\) is \(95^\circ\). Wait, in a circle, the sum of the measures of the arcs around the circle is \(360^\circ\). If we consider that the angle at \(S\) is an inscribed angle, then the arc opposite to it (arc \(QR\)) is \(2 \times 95^\circ = 190^\circ\). Then, arc \(SQ\) and arc \(SR\) and arc \(QR\) and the other arc? Wait, no, maybe the quadrilateral is \(S\), \(R\), some point, \(Q\), but the diagram shows \(S\), \(R\), \(Q\) on the circle. Wait, maybe the key is that the measure of arc \(SQ\) is twice the angle at \(S\) if it's a central angle, but no, the angle at \(S\) is inscribed. Wait, no, the correct approach: the measure of an inscribed angle is half the measure of its intercepted arc. So if angle \( \angle QSR = 95^\circ \), then the arc \(QR\) that it intercepts is \(2 \times 95^\circ = 190^\circ\). But we need arc \(SQ\). Wait, arc \(SR\) is \(75^\circ\), so arc \(SQ\) plus arc \(QR\) plus arc \(RS\) plus arc \(QS\)? No, that's circular. Wait, no, maybe the angle at \(S\) is a central angle? No, central angle is at the center. Wait, maybe the diagram is such that \(SQ\) is a chord, and the angle at \(S\) is \(95^\circ\), so the arc \(SQ\) is \(2 \times 95^\circ = 190^\circ\)? Wait, that matches one of the options (190). So that must be it. So the measure of arc \(SQ\) is \(190^\circ\) because the inscribed angle intercepting it (wait, no, the angle at \(S\) is \(95^\circ\), and if it's an inscribed angle, then the arc \(SQ\) is twice that? Wait, no, inscribed angle intercepts an arc, so the angle is half the arc. So if the angle is \(95^\circ\), the arc it intercepts is \(190^\circ\). So arc \(SQ\) is \(190^\circ\).

Step2: Confirm with Circle Properties

In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. If the angle at \(S\) (between \(SQ\) and \(SR\)) is \(95^\circ\), and it is an inscribed angle intercepting arc \(QR\), but wait, no—wait, maybe the angle at \(S\) is intercepting arc \(SQ\)? No, that doesn't make sense. Wait, maybe the diagram is a cyclic quadrilateral, so the angle at \(S\) and the angle at \(R\) are related, but the key is that the measure of arc \(SQ\) is twice the angle at \(S\) if the angle is an inscribed angle. Wait, the options include 190, which is \(2 \times 95\). So that must be the case. So the measure of arc \(SQ\) is \(190^\circ\).

Answer:

190