1. **Problem Statement:** Given a circle with center $O$ and diameter $AB$, and points $S$ and $T$ on the circle, we need to find the measures of arcs $AST$, $SB$, $BT$, and $ATB$ given that $m\angle AOT = 44^\circ$. 2. **Key Concepts:** - The measure of a central angle equals the measure of its intercepted arc. - Since $AB$ is a diameter, it subtends a $180^\circ$ arc. - The circle's total circumference is $360^\circ$. 3. **Step-by-step Solution:** - $m\angle AOT = 44^\circ$ means the arc $AT$ measures $44^\circ$. - Since $AB$ is a diameter, arc $AB$ measures $180^\circ$. - Arc $AST$ includes arcs $AT$ and $ST$. To find $m(\text{arc } AST)$, we need $m(\text{arc } ST)$. - Angle $SOT$ is the central angle intercepting arc $ST$. Since $O$ is center, $m(\text{arc } ST) = m\angle SOT$. - But $m\angle AOT = 44^\circ$ and $A$, $S$, $T$ are points on the circle, so $m\angle SOT$ can be found if given or deduced. Since not given, assume $m\angle SOT = 44^\circ$ (if $S$ lies between $A$ and $T$ on the circle). - Then $m(\text{arc } AST) = m(\text{arc } AS) + m(\text{arc } ST) = 44^\circ + 44^\circ = 88^\circ$. - Arc $SB$ is the remaining part of the semicircle from $S$ to $B$ on the diameter side. Since $AB=180^\circ$, and $AS + SB = 180^\circ$, then $m(\text{arc } SB) = 180^\circ - m(\text{arc } AS)$. - If $m(\text{arc } AS) = 44^\circ$, then $m(\text{arc } SB) = 180^\circ - 44^\circ = 136^\circ$. - Arc $BT$ is the minor arc from $B$ to $T$ on the circle. Since $m(\text{arc } AT) = 44^\circ$ and $AB=180^\circ$, then $m(\text{arc } BT) = 360^\circ - 180^\circ - 44^\circ = 136^\circ$. - Arc $ATB$ is the major arc passing through $A$, $T$, and $B$. Since minor arc $AB=180^\circ$, and minor arc $AT=44^\circ$, then $m(\text{arc } ATB) = 360^\circ - 44^\circ = 316^\circ$. 4. **Final answers:** - $m(\text{arc } AST) = 88^\circ$ - $m(\text{arc } SB) = 136^\circ$ - $m(\text{arc } BT) = 136^\circ$ - $m(\text{arc } ATB) = 316^\circ$