Question

1. in circle r with ( mangle qts = 33^{circ} ), find the ( mangle qrs ).
2. if ( mwidehat{gu}=42^{circ} ) and ( mangle a = 44^{circ} ), find ( mwidehat{li} ).

Explanation:

Step1: Use the inscribed - angle theorem

The inscribed - angle theorem states that the measure of an inscribed angle is half the measure of the central angle that subtends the same arc. In circle \(R\), \(\angle QTS\) is an inscribed angle and \(\angle QRS\) is the central angle subtending the same arc \(\overset{\frown}{QS}\).
Let \(m\angle QTS=\theta = 33^{\circ}\) and \(m\angle QRS=\alpha\). According to the inscribed - angle theorem \(\theta=\frac{1}{2}\alpha\).

Step2: Solve for the central angle

From \(\theta=\frac{1}{2}\alpha\), we can solve for \(\alpha\) by multiplying both sides of the equation by \(2\). So \(\alpha = 2\theta\).
Substitute \(\theta = 33^{\circ}\) into the equation: \(\alpha=2\times33^{\circ}\).

Answer:

\(m\angle QRS = 66^{\circ}\)