Question

what is the measure of arc qr? 26° 52° 104° 128°

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The formula is \(m\angle=\frac{1}{2}m\overset{\frown}{arc}\), where \(m\angle\) is the measure of the inscribed angle and \(m\overset{\frown}{arc}\) is the measure of the intercepted arc.

Step2: Identify the inscribed angle and its intercepted arc

In the given circle, \(\angle QSR\) is an inscribed angle and \(\overset{\frown}{QR}\) is the intercepted arc. We know that the diameter \(QS\) forms a straight - line (a \(180^{\circ}\) angle at the center). The inscribed angle \(\angle R = 52^{\circ}\).
By the inscribed - angle theorem, if \(\angle R\) is an inscribed angle and \(\overset{\frown}{QR}\) is the intercepted arc, then \(m\angle R=\frac{1}{2}m\overset{\frown}{QR}\). But wait, another way: The central angle corresponding to the arc \(\overset{\frown}{QR}\) can be found as follows.
Since \(QS\) is a diameter (\(180^{\circ}\)), and if we consider the relationship between the inscribed angle and the arc. The inscribed angle \(\angle R = 52^{\circ}\), but actually, if we use the property that the measure of an arc is twice the measure of an inscribed angle that intercepts it.
Let \(x\) be the measure of arc \(\overset{\frown}{QR}\).
We know that the inscribed angle \(\angle R\) (wait, no, \(\angle R\) is not the inscribed angle for \(\overset{\frown}{QR}\) in the wrong sense. The correct approach: The inscribed angle \(\angle R\) (assuming the standard circle - angle relations). Wait, actually, if we consider the central - angle formula.
The measure of an arc is equal to the measure of its central angle. If we use the property that the inscribed angle \(\angle R\) (wait, no, \(\angle R\) is not the right one. Wait, the inscribed angle that intercepts \(\overset{\frown}{QR}\) is \(\angle R\) (assuming the circle with center \(T\)).
By the inscribed - angle theorem \(m\overset{\frown}{QR}=2\times m\angle R\). But wait, no, if we consider the fact that \(QS\) is a diameter (\(180^{\circ}\)).
Another approach: The measure of an arc \(\overset{\frown}{QR}\) is related to the inscribed angle. The inscribed angle \(\angle R\) (assuming the correct interception). The measure of an inscribed angle \(\theta\) that intercepts an arc \(A\) is given by \(\theta=\frac{1}{2}A\).
If we assume that the given \(52^{\circ}\) is the inscribed angle for arc \(\overset{\frown}{QR}\), then \(m\overset{\frown}{QR}=2\times52^{\circ}\)

Step3: Calculate the measure of arc \(\overset{\frown}{QR}\)

$$m\overset{\frown}{QR}=2\times52^{\circ}=104^{\circ}$$

Answer:

\(104^{\circ}\)