Angles Inside Circle F6509B

1. **State the problem:** We need to find the values of the unknown angles $q$ and $s$ inside a circle with given arcs of $23^\circ$, $127^\circ$, and $48^\circ$. 2. **Recall the rule for angles formed by intersecting chords inside a circle:** The measure of an angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 3. **Identify the arcs intercepted by angles $q$ and $s$:** - Angle $q$ intercepts arcs $23^\circ$ and $48^\circ$. - Angle $s$ intercepts arcs $127^\circ$ and the arc opposite to $s$, which is the sum of $23^\circ$ and $48^\circ$ (since the full circle is $360^\circ$, the remaining arc is $360^\circ - 127^\circ = 233^\circ$, but here we use the arcs adjacent to $s$). 4. **Calculate $q$:** $$q = \frac{23^\circ + 48^\circ}{2} = \frac{71^\circ}{2} = 35.5^\circ$$ 5. **Calculate $s$:** $$s = \frac{127^\circ + 48^\circ}{2} = \frac{175^\circ}{2} = 87.5^\circ$$ 6. **Check the options:** None of the options list $s = 87.5^\circ$, so re-examine the arcs for $s$. 7. **Reconsider arcs for $s$:** Since $q$ and $s$ are vertical angles, $s$ intercepts arcs $127^\circ$ and $23^\circ$. 8. **Recalculate $s$:** $$s = \frac{127^\circ + 23^\circ}{2} = \frac{150^\circ}{2} = 75^\circ$$ 9. **Sum of $q$ and $s$:** $$35.5^\circ + 75^\circ = 110.5^\circ$$ which does not match any given options. 10. **Alternative approach:** The sum of arcs around the circle is $360^\circ$. Given arcs are $23^\circ$, $127^\circ$, and $48^\circ$, sum is $198^\circ$. The remaining arc is: $$360^\circ - 198^\circ = 162^\circ$$ 11. **Assuming $s$ intercepts arcs $127^\circ$ and $162^\circ$:** $$s = \frac{127^\circ + 162^\circ}{2} = \frac{289^\circ}{2} = 144.5^\circ$$ 12. **$q$ intercepts arcs $23^\circ$ and $48^\circ$:** $$q = \frac{23^\circ + 48^\circ}{2} = 35.5^\circ$$ 13. **Final answer:** $$q = 35.5^\circ, \quad s = 144.5^\circ$$ This matches the first option. **Answer:** $q = 35.5$ and $s = 144.5$.