Question

what is ( mangle qsr )?

Explanation:

Step1: Use the central - angle and inscribed - angle relationship

The measure of an inscribed angle is half of the measure of the central angle that subtends the same arc. Here, \(\angle QSR\) is an inscribed angle and \(\angle QPR\) is the central angle subtending the same arc \(QR\). So, \(m\angle QPR = 2m\angle QSR\).
Given \(m\angle QSR=x - 46^{\circ}\) and \(m\angle QPR=x - 9^{\circ}\), we have the equation \(x-9^{\circ}=2(x - 46^{\circ})\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(x-9^{\circ}=2x-92^{\circ}\).
Subtract \(x\) from both sides: \(-9^{\circ}=2x - x-92^{\circ}\).
Simplify: \(-9^{\circ}=x - 92^{\circ}\).
Add \(92^{\circ}\) to both sides: \(x=-9^{\circ}+92^{\circ}=83^{\circ}\).

Step3: Find \(m\angle QSR\)

Substitute \(x = 83^{\circ}\) into the formula for \(m\angle QSR\). Since \(m\angle QSR=x - 46^{\circ}\), then \(m\angle QSR=83^{\circ}-46^{\circ}\).
\(m\angle QSR = 37^{\circ}\)

Answer:

\(37\)