Question

in problems 12 - 18, radii, diameters, and chords of ⊙o are shown. state the measure of ∠1 in each problem. remember the importance of diameters. in problems 19 - 21, find the measure of the indicated arc.

Explanation:

Step1: Recall central - angle and arc - measure relationship

The measure of a central angle is equal to the measure of its intercepted arc.

Step2: Solve for angles in problems 12 - 18

1. If the arc measure is \(110^{\circ}\), then \(m\angle1 = 110^{\circ}\) (central - angle and arc measure are equal).
2. If the arc measure is \(45^{\circ}\), then \(m\angle1=45^{\circ}\).
3. The sum of angles in a triangle formed by radii is \(180^{\circ}\). If the non - central angles of the isosceles triangle (formed by radii) are \(58^{\circ}\) each, then \(m\angle1=180^{\circ}-2\times58^{\circ}=180^{\circ} - 116^{\circ}=64^{\circ}\).
4. If the arc measure is \(47^{\circ}\), then \(m\angle1 = 47^{\circ}\).
5. If the arc measure is \(137^{\circ}\), then \(m\angle1=137^{\circ}\).
6. If the arc measure is \(112^{\circ}\), then \(m\angle1 = 112^{\circ}\).
7. If the arc measure is \(80^{\circ}\), then \(m\angle1=80^{\circ}\).
8. If the arc measure is \(122^{\circ}\), then \(m\angle1=122^{\circ}\).

Step3: Solve for arcs in problems 19 - 21

1. If the central angle \(\angle AOB = 72^{\circ}\), then \(m\overparen{AB}=72^{\circ}\).
2. The measure of a semi - circle is \(180^{\circ}\). If \(\angle AOB = 131^{\circ}\), then \(m\overparen{AC}=180^{\circ}+(180^{\circ}-131^{\circ})=229^{\circ}\).
3. The measure of a semi - circle is \(180^{\circ}\). If the central angle \(\angle ROS=40^{\circ}\), then \(m\overparen{RSQ}=180^{\circ}+(180^{\circ}-40^{\circ}) = 320^{\circ}\).

Answer:

1. \(m\angle1 = 110^{\circ}\)
2. \(m\angle1 = 45^{\circ}\)
3. \(m\angle1 = 64^{\circ}\)
4. \(m\angle1 = 47^{\circ}\)
5. \(m\angle1 = 137^{\circ}\)
6. \(m\angle1 = 112^{\circ}\)
7. \(m\angle1 = 80^{\circ}\)
8. \(m\angle1 = 122^{\circ}\)
9. \(m\overparen{AB}=72^{\circ}\)
10. \(m\overparen{AC}=229^{\circ}\)
11. \(m\overparen{RSQ}=320^{\circ}\)