1. **Problem A: Find the measure of each arc of circle ⊙P.** Given: Circle with center P, points A, B, C on circumference. Angle at center P between radii PC and PB is 57°. 2. **Recall the rule:** The measure of a central angle equals the measure of its intercepted arc. 3. **Find each arc:** - $\overset{\frown}{CB}$: Since $\angle CPB = 57^\circ$, arc $CB$ measures $57^\circ$. - $\overset{\frown}{BCA}$: This arc is the major arc from B to A passing through C. The full circle is $360^\circ$, so subtract the minor arc $BA$ (to be found next). - $\overset{\frown}{BA}$: Since $\angle BPA$ is not given, but $\angle CPB = 57^\circ$ and points A, B, C are on the circle, assume arcs $BA$ and $CB$ are complementary to $57^\circ$. - $\overset{\frown}{ABC}$: This is the arc opposite to $CB$, so $360^\circ - 57^\circ = 303^\circ$. 4. **Summary for A:** - $\overset{\frown}{CB} = 57^\circ$ - $\overset{\frown}{ABC} = 303^\circ$ - $\overset{\frown}{BA}$ and $\overset{\frown}{BCA}$ require more info or assumptions; with given data, only $\overset{\frown}{CB}$ and $\overset{\frown}{ABC}$ can be confidently stated. 5. **Problem B: Find the measure of each arc of circle ⊙D.** Given: Circle with center D, points F, G, H, E on circumference. Central angles: - $\angle FDG = 113^\circ$ - $\angle GDE = 92^\circ$ - $\angle HDG = 43^\circ$ 6. **Recall:** The measure of a central angle equals the measure of its intercepted arc. 7. **Find arcs:** - $\overset{\frown}{FH}$: This arc is the sum of arcs $FG$ and $GH$. - $\overset{\frown}{EFH}$: This is the major arc passing through E, F, H. - $\overset{\frown}{HE}$: Arc from H to E. 8. **Calculate arcs:** - $\overset{\frown}{FG} = 113^\circ$ - $\overset{\frown}{GE} = 92^\circ$ - $\overset{\frown}{HG} = 43^\circ$ - $\overset{\frown}{FH} = \overset{\frown}{FG} + \overset{\frown}{GH} = 113^\circ + 43^\circ = 156^\circ$ - $\overset{\frown}{HE} = 360^\circ - (\overset{\frown}{FH} + \overset{\frown}{EF}) = 360^\circ - (156^\circ + 92^\circ) = 112^\circ$ - $\overset{\frown}{EFH}$ is the major arc passing through E, F, H, so $\overset{\frown}{EFH} = 360^\circ - \overset{\frown}{GH} = 360^\circ - 43^\circ = 317^\circ$ 9. **Problem C: Determine if arcs are congruent and justify.** - For arcs $KM$ and $NJ$ with central angles $63^\circ$ each, arcs are congruent by the theorem: "Arcs intercepted by congruent central angles are congruent." - For arcs $LY$ and $RX$ with central angles $72^\circ$ each, arcs are congruent by the same theorem. - For arcs $MN$ and $PO$ in concentric circles, since arc $MP$ is $46^\circ$, and arcs $MN$ and $PO$ are highlighted, if their central angles are equal, arcs are congruent by the same theorem. Without central angle info, cannot confirm. **Final answers:** - A1. $\overset{\frown}{CB} = 57^\circ$ - A3. $\overset{\frown}{ABC} = 303^\circ$ - B5. $\overset{\frown}{FH} = 156^\circ$ - B6. $\overset{\frown}{EFH} = 317^\circ$ - B7. $\overset{\frown}{HE} = 112^\circ$ - C8. Arcs $KM$ and $NJ$ are congruent by congruent central angles. - C9. Arcs $LY$ and $RX$ are congruent by congruent central angles. - C10. Insufficient info to confirm congruency.