Question

relating angles of a circle. which statements are true? check all that apply. the circumscribed angle l has a measure of 49°. the circumscribed angle l has a measure of 65.5°. if the measure of arc hj is 98°, the measure of angle hmj is 98°. if the measure of arc hj is 98°, the measure of angle hkj is 98°. if the measure of arc hj is 98°, the measure of arc hk is 131°.

Explanation:

Step1: Recall circle - angle relationships

The measure of a central angle is equal to the measure of its intercepted arc. The measure of an inscribed - angle is half the measure of its intercepted arc. The measure of a circumscribed angle is supplementary to the central angle that intercepts the same arc.
Let the central angle \(\angle HMJ\) intercept arc \(HJ\). If the measure of arc \(HJ\) is \(98^{\circ}\), then the measure of \(\angle HMJ\) is \(98^{\circ}\) (central - angle rule).
Let the inscribed angle \(\angle HKJ\) intercept arc \(HJ\). Then the measure of \(\angle HKJ=\frac{1}{2}\text{ measure of arc }HJ\). So if arc \(HJ = 98^{\circ}\), then \(\angle HKJ = 49^{\circ}\).
Let the circumscribed angle \(\angle L\) intercept arc \(HJ\). The central angle \(\angle HMJ\) and the circumscribed angle \(\angle L\) are supplementary. If \(\angle HMJ = 131^{\circ}\), then \(\angle L=180 - 131=49^{\circ}\).

Step2: Analyze each statement

1. For the measure of the circumscribed angle \(L\):

• Since the central angle \(\angle HMJ = 131^{\circ}\), and the circumscribed angle \(L\) and the central angle \(\angle HMJ\) are supplementary (\(\angle L+\angle HMJ = 180^{\circ}\)), \(\angle L=180 - 131 = 49^{\circ}\). So the statement "The circumscribed angle \(L\) has a measure of \(49^{\circ}\)" is true.

1. "The circumscribed angle \(L\) has a measure of \(65.5^{\circ}\)" is false as we found \(\angle L = 49^{\circ}\).
2. "If the measure of arc \(HJ\) is \(98^{\circ}\), the measure of angle \(HMJ\) is \(98^{\circ}\)" is true because the measure of a central angle is equal to the measure of its intercepted arc.
3. "If the measure of arc \(HJ\) is \(98^{\circ}\), the measure of angle \(HKJ\) is \(98^{\circ}\)" is false. Since \(\angle HKJ\) is an inscribed angle intercepting arc \(HJ\), \(\angle HKJ=\frac{1}{2}\times98^{\circ}=49^{\circ}\).
4. "If the measure of arc \(HJ\) is \(98^{\circ}\), the measure of arc \(HK\) is \(131^{\circ}\)" is false. There is no information given to support this relationship between arc \(HJ\) and arc \(HK\) based on the problem - setup.

Answer:

The circumscribed angle \(L\) has a measure of \(49^{\circ}\), If the measure of arc \(HJ\) is \(98^{\circ}\), the measure of angle \(HMJ\) is \(98^{\circ}\)