Question

in the diagram of circle a, what is ( mangle lmn )? ( 75^{circ} ) ( 90^{circ} ) ( 120^{circ} ) ( 135^{circ} )

Explanation:

Step1: Find the measure of the minor arc

The total degrees in a circle is \(360^{\circ}\). Given the major arc \(LN = 270^{\circ}\), so the minor arc \(LN=360^{\circ}- 270^{\circ}=90^{\circ}\)

Step2: Use the formula for the measure of an angle formed by two tangents

The measure of an angle formed by two tangents outside a circle is half the difference of the measures of the intercepted arcs. Here, the formula is \(m\angle LMN=\frac{1}{2}(major\ arc\ LN - minor\ arc\ LN)\)
Substitute the values: \(m\angle LMN=\frac{1}{2}(270^{\circ}-90^{\circ})\)

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Answer:

\(90^{\circ}\) (the second option)