Question

the measure of arc lm is \boxed{}°.
the measure of angle mbl is \boxed{}°.
the measure of angle mnl is \boxed{}°.
(there is a circle with center o, points b, m, l on the circle, angle lom is 116°, and point n outside the circle connected to m and l.)

Explanation:

For the measure of arc \( LM \):

Step 1: Recall central angle - arc relation

The central angle \( \angle LOM \) is given as \( 116^\circ \). The measure of an arc is equal to the measure of its central angle.
\( \text{Measure of arc } LM = \text{Measure of } \angle LOM \)

Step 2: Determine the measure

Since \( \angle LOM = 116^\circ \), the measure of arc \( LM \) is \( 116^\circ \).

For the measure of angle \( MBL \):

Step 1: Recall inscribed angle theorem

The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Angle \( \angle MBL \) intercepts arc \( LM \).
\( \angle MBL=\frac{1}{2}\times \text{Measure of arc } LM \)

Step 2: Substitute the arc measure

We know arc \( LM = 116^\circ \), so \( \angle MBL=\frac{1}{2}\times116^\circ = 58^\circ \).

For the measure of angle \( MNL \):

Step 1: Recall tangent - chord angle theorem

The angle between a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. \( LN \) is a tangent to the circle at \( L \), and \( LM \) is a chord. So \( \angle MNL \) is equal to the inscribed angle \( \angle MBL \) (or half the measure of arc \( LM \)).
\( \angle MNL=\frac{1}{2}\times \text{Measure of arc } LM \)

Step 2: Substitute the arc measure

Since arc \( LM = 116^\circ \), \( \angle MNL=\frac{1}{2}\times116^\circ= 58^\circ \).

Answer:

s:

• The measure of arc \( LM \) is \( \boldsymbol{116^\circ} \).
• The measure of angle \( MBL \) is \( \boldsymbol{58^\circ} \).
• The measure of angle \( MNL \) is \( \boldsymbol{58^\circ} \).