Question

the measure of arc lm is
the measure of angle mbl is
the measure of angle mnl is
58
64
116
180

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of an arc is equal to the measure of its central angle. Given that the central angle $\angle LOM = 116^{\circ}$, the measure of arc $LM$ is $116^{\circ}$.

Step2: Recall inscribed - central angle relationship for $\angle MBL$

The measure of an inscribed angle is half of the measure of the central angle subtended by the same arc. For inscribed angle $\angle MBL$ and central angle $\angle MOL$ subtended by arc $LM$, $\angle MBL=\frac{1}{2}\angle MOL$. So $\angle MBL = \frac{1}{2}\times116^{\circ}=58^{\circ}$.

Step3: Recall inscribed - central angle relationship for $\angle MNL$

The measure of an inscribed angle is half of the measure of the central angle subtended by the same arc. For inscribed angle $\angle MNL$ and central angle $\angle MOL$ subtended by arc $LM$, $\angle MNL=\frac{1}{2}\angle MOL$. So $\angle MNL = \frac{1}{2}\times116^{\circ}=58^{\circ}$.

Answer:

The measure of arc $LM$ is $116^{\circ}$.
The measure of angle $MBL$ is $58^{\circ}$.
The measure of angle $MNL$ is $58^{\circ}$.