Question

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 a circum - scribed angle is supplementary to the central angle that intercepts the same arc.

Step2: Analyze the circum - scribed angle

Let the central angle corresponding to arc HJ be $\angle HMJ$. If $\angle HMJ = 131^{\circ}$, and the circum - scribed angle $\angle L$ and the central angle $\angle HMJ$ are related as $\angle L=180^{\circ}-\angle HMJ$. So $\angle L = 180 - 131=49^{\circ}$.

Step3: Central angle and arc relationship

The measure of a central angle is equal to the measure of its intercepted arc. So if the measure of arc HJ is $98^{\circ}$, the measure of $\angle HMJ$ (central angle) is $98^{\circ}$.

Step4: Inscribed - angle and arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc. If the measure of arc HJ is $98^{\circ}$, the measure of inscribed angle $\angle HKJ=\frac{1}{2}\times98^{\circ} = 49^{\circ}$, not $98^{\circ}$. And there is no information to support the relationship between arc HJ and arc HK as described in the last option.

Answer:

The circumscribed angle L has a measure of 49°.
If the measure of arc HJ is 98°, the measure of angle HMJ is 98°.