Question

applying the radius-tangent theorem\
ewline $\overline{lm}$ is tangent to $\odot n$ at point $m$.\
ewline determine the following angle measures.\
ewline $m\angle m = \square^{\circ}$\
ewline $m\angle l = \square^{\circ}$

Explanation:

Step1: Find \( m\angle M \)

The Radius - Tangent Theorem states that a radius is perpendicular to a tangent at the point of tangency. So, \( \overline{NM} \perp \overline{LM} \), which means \( \angle M = 90^{\circ} \).

Step2: Find \( m\angle L \)

In triangle \( NML \), we know that the sum of the interior angles of a triangle is \( 180^{\circ} \). We already know that \( \angle M = 90^{\circ} \) and \( \angle N=66^{\circ} \). Let \( \angle L=x \). Then, by the angle - sum property of a triangle: \( x + 90^{\circ}+66^{\circ}=180^{\circ} \).
Simplify the equation: \( x+156^{\circ}=180^{\circ} \).
Subtract \( 156^{\circ} \) from both sides: \( x = 180^{\circ}- 156^{\circ}=24^{\circ} \). So, \( m\angle L = 24^{\circ} \).

Answer:

\( m\angle M=\boldsymbol{90} \), \( m\angle L=\boldsymbol{24} \)