1. **Problem:** In circle P, given $m\overset{\frown}{QR} = 110^\circ$, $m\overset{\frown}{RS} = 94^\circ$, and $m\angle QRT = 27^\circ$, find the measures requested. 2. **Formula and rules:** - The measure of an inscribed angle is half the measure of its intercepted arc: $$m\angle = \frac{1}{2} m\overset{\frown}{arc}$$ - The sum of arcs around a circle is $360^\circ$. - Angles on a straight line sum to $180^\circ$. 3. **Given:** - $m\overset{\frown}{QR} = 110^\circ$ - $m\overset{\frown}{RS} = 94^\circ$ - $m\angle QRT = 27^\circ$ - $m\angle QTR = 63^\circ$ (given) - $mQT = 54^\circ$ (given) 4. **Find:** - b) $m\angle RQS$ - c) $mTS$ - d) $m\angle TRS$ - e) $m\angle QSR$ --- **Step b) Find $m\angle RQS$** - $m\angle RQS$ intercepts arc $RT$. - Arc $RT = m\overset{\frown}{QR} + m\overset{\frown}{RS} = 110^\circ + 94^\circ = 204^\circ$ - Using inscribed angle formula: $$m\angle RQS = \frac{1}{2} \times 204^\circ = 102^\circ$$ **Step c) Find $mTS$ (arc measure)** - The total circle is $360^\circ$. - Known arcs: $m\overset{\frown}{QR} = 110^\circ$, $m\overset{\frown}{RS} = 94^\circ$, $m\overset{\frown}{QT} = 54^\circ$ (given) - Arc $TS = 360^\circ - (m\overset{\frown}{QR} + m\overset{\frown}{RS} + m\overset{\frown}{QT})$ - Calculate: $$m\overset{\frown}{TS} = 360^\circ - (110^\circ + 94^\circ + 54^\circ) = 360^\circ - 258^\circ = 102^\circ$$ **Step d) Find $m\angle TRS$** - $m\angle TRS$ intercepts arc $TS$. - Using inscribed angle formula: $$m\angle TRS = \frac{1}{2} \times 102^\circ = 51^\circ$$ **Step e) Find $m\angle QSR$** - $m\angle QSR$ intercepts arc $QR$. - Using inscribed angle formula: $$m\angle QSR = \frac{1}{2} \times 110^\circ = 55^\circ$$ --- **Final answers:** - a) $m\angle QTR = 63^\circ$ (given) - b) $m\angle RQS = 102^\circ$ - c) $mTS = 102^\circ$ - d) $m\angle TRS = 51^\circ$ - e) $m\angle QSR = 55^\circ$ - f) $mQT = 54^\circ$ (given)