Circle Angles B09F61

1. **State the problem:** We have a circle with two intersecting chords creating angles $a$ and $b$ inside the circle. Given arcs of $37^\circ$, $164^\circ$, and $45^\circ$, we want to find the values of $a$ and $b$. 2. **Recall the formula for angles formed by intersecting chords inside a circle:** $$a = \frac{1}{2}(\text{sum of opposite arcs})$$ Specifically, the 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 for angles $a$ and $b$:** - For angle $a$, the intercepted arcs are $37^\circ$ and $41^\circ$ (since $a$ is near the arc labeled $45^\circ$, and $41^\circ$ is the complementary arc to $45^\circ$ in the circle). - For angle $b$, the intercepted arcs are $164^\circ$ and $118^\circ$ (since $b$ is near the arcs $164^\circ$ and the remaining arc $118^\circ$ to complete $360^\circ$). 4. **Calculate $a$:** $$a = \frac{1}{2}(37 + 41) = \frac{1}{2}(78) = 39^\circ$$ 5. **Calculate $b$:** $$b = \frac{1}{2}(164 + 118) = \frac{1}{2}(282) = 141^\circ$$ 6. **Conclusion:** The values are $a = 39^\circ$ and $b = 141^\circ$. This matches the option: $b = 141$ and $a = 39$.