1. **State the problem:** We are given a circle with points P, Q, R, S on the circumference and angles 126°, 93°, and 90° at certain points. We need to find the measures of angles $\angle QRS$, $\angle PSR$, $\angle QPS$, and the measures of arcs $QRS$ and $SPQ$. 2. **Given information:** - $m\angle PQS = 126^\circ$ (angle at arc PQ) - $m\angle QRS = 93^\circ$ - $m\angle PS = 90^\circ$ 3. **Recall important rules:** - The measure of an inscribed angle is half the measure of its intercepted arc. - Opposite angles of a cyclic quadrilateral sum to $180^\circ$. 4. **Find $m\angle QRS$:** Given as $93^\circ$. 5. **Find $m\angle PSR$:** Since $\angle PSR$ and $\angle QRS$ subtend the same arc $PR$, and $\angle QRS = 93^\circ$, then $m\angle PSR = 93^\circ$. 6. **Find $m\angle QPS$:** Since $\angle QPS$ is an inscribed angle intercepting arc $QS$, and $m\angle PQS = 126^\circ$ intercepts arc $PS$, use the cyclic quadrilateral property: $$m\angle QPS + m\angle QRS = 180^\circ$$ $$m\angle QPS + 93^\circ = 180^\circ$$ $$m\angle QPS = 180^\circ - 93^\circ = 87^\circ$$ 7. **Find measure of arc $QRS$:** The inscribed angle $\angle QPS = 87^\circ$ intercepts arc $QRS$, so $$m\text{arc } QRS = 2 \times 87^\circ = 174^\circ$$ 8. **Find measure of arc $SPQ$:** Since the total circle is $360^\circ$, $$m\text{arc } SPQ = 360^\circ - 174^\circ = 186^\circ$$ **Final answers:** - $m\angle QRS = 93^\circ$ - $m\angle PSR = 93^\circ$ - $m\angle QPS = 87^\circ$ - $m\text{arc } QRS = 174^\circ$ - $m\text{arc } SPQ = 186^\circ$