Question

in the diagram of circle a, what is m∠lmn? 75° 90° 120° 135°

Explanation:

Step1: Recall the formula for the measure of an angle formed by two tangents

The measure of an angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs.
Let the major - arc be $m\overset{\frown}{LN}=270^{\circ}$ and the minor - arc be $m\overset{\frown}{LN}=360^{\circ}-270^{\circ} = 90^{\circ}$.
The formula for $\angle LMN$ (angle formed by two tangents $ML$ and $MN$) is $m\angle LMN=\frac{1}{2}(m\overset{\frown}{LN_{major}}-m\overset{\frown}{LN_{minor}})$.

Step2: Substitute the values of the arcs

Substitute $m\overset{\frown}{LN_{major}} = 270^{\circ}$ and $m\overset{\frown}{LN_{minor}}=90^{\circ}$ into the formula:
$m\angle LMN=\frac{1}{2}(270 - 90)$.
First, calculate the value inside the parentheses: $270-90 = 180$.
Then, $\frac{1}{2}\times180=90^{\circ}$.

Answer:

$90^{\circ}$