Question

what is ( mangle qrs )?
( mangle qrs=square^{circ} )

Explanation:

Step1: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc. So, \(m\angle QRS=\frac{1}{2}m\angle QTS\). Given \(m\angle QRS = x - 2\) and \(m\angle QTS=x + 63\), we have the equation \(x-2=\frac{1}{2}(x + 63)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(x-2=\frac{1}{2}(x + 63)\) by \(2\):
\(2(x - 2)=x + 63\)
Expand the left - hand side: \(2x-4=x + 63\)
Subtract \(x\) from both sides: \(2x-x-4=x - x+ 63\), which gives \(x-4 = 63\)
Add \(4\) to both sides: \(x=63 + 4=67\)

Step3: Find \(m\angle QRS\)

Substitute \(x = 67\) into the formula for \(m\angle QRS\). Since \(m\angle QRS=x - 2\), then \(m\angle QRS=67-2\)

Answer:

\(65\)