\( \triangle L M N, \angle N \) is a right angle, \( L M=76 \) units, and \( M N=40 \) units. What is the approximate measure of \( \angle M \) ? A. \( 58^{\circ} \) B. \( 28^{\circ} \) C. \( 32^{\circ} \) D. \( 40^{\circ} \)

To find the measure of \( \angle M \) in triangle \( \triangle LMN \) where \( \angle N \) is a right angle, we can use the tangent function. Since we know the lengths of the sides \( LM \) (the hypotenuse) and \( MN \) (the opposite side for \( \angle M \)), we can express it as: \[ \tan(M) = \frac{MN}{LN} \] First, we need to find the length of \( LN \) using the Pythagorean theorem: \[ LN = \sqrt{LM^2 - MN^2} = \sqrt{76^2 - 40^2} = \sqrt{5776 - 1600} = \sqrt{4176} \approx 64.5 \text{ units} \] Then we can calculate \( \tan(M) \): \[ \tan(M) = \frac{40}{64.5} \approx 0.6207 \] Using the arctangent function to find \( \angle M \): \[ M \approx \tan^{-1}(0.6207) \approx 31.7^{\circ} \] Rounding gives us approximately \( 32^{\circ} \). Thus, the answer is: C. \( 32^{\circ} \)