Question

use the diagram to complete the statements. the measure of angle l is \boxed{}°. the trigonometric ratio that uses ∠m and ln to solve for nm is \boxed{}. the length of nm, to the nearest tenth, is approximately \boxed{}. diagram: right triangle lnm with right angle at n, ln = 21, angle at m is 20°.

Explanation:

Step1: Find measure of angle L

In a right - triangle \( \triangle LNM \) with \( \angle N = 90^{\circ} \) and \( \angle M=20^{\circ} \). We know that the sum of angles in a triangle is \( 180^{\circ} \). So, \( \angle L+\angle N+\angle M = 180^{\circ} \). Substituting \( \angle N = 90^{\circ} \) and \( \angle M = 20^{\circ} \), we get \( \angle L+90^{\circ}+20^{\circ}=180^{\circ} \). Then \( \angle L=180^{\circ}-(90^{\circ} + 20^{\circ})=70^{\circ} \).

Step2: Identify trigonometric ratio

We know that \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \angle M \), the opposite side to \( \angle M \) is \( LN \) and the adjacent side is \( NM \). So \( \tan M=\frac{LN}{NM} \), which can be rearranged to solve for \( NM \) as \( NM=\frac{LN}{\tan M} \). So the trigonometric ratio is tangent.

Step3: Calculate length of NM

We know that \( LN = 21 \) and \( \angle M = 20^{\circ} \). Using the formula \( NM=\frac{LN}{\tan M} \), and \( \tan(20^{\circ})\approx0.3640 \). Then \( NM=\frac{21}{\tan(20^{\circ})}=\frac{21}{0.3640}\approx57.7 \)? Wait, no, wait. Wait, if we consider \( \angle L = 70^{\circ} \), maybe I mixed up. Wait, let's re - examine. Wait, \( \angle L = 70^{\circ} \), \( \angle M=20^{\circ} \), \( LN = 21 \) (opposite to \( \angle M \)), \( NM \) is adjacent to \( \angle M \). Wait, \( \tan M=\frac{LN}{NM}\Rightarrow NM=\frac{LN}{\tan M} \). \( \tan(20^{\circ})\approx0.3640 \), \( LN = 21 \), so \( NM=\frac{21}{0.3640}\approx57.7 \)? But the options given are 20, 70, 90? Wait, maybe I made a mistake. Wait, maybe \( \angle L = 70^{\circ} \), and we can use \( \tan L=\frac{NM}{LN} \). Since \( \angle L = 70^{\circ} \), \( \tan(70^{\circ})\approx2.7475 \). Then \( NM = LN\times\tan L=21\times2.7475\approx57.7 \)? But the options are 20, 70, 90? Wait, maybe the options are for the angle first. Wait, the first answer is \( 70^{\circ} \), the trigonometric ratio is tangent, and for \( NM \), let's recalculate. Wait, if \( \angle M = 20^{\circ} \), \( LN = 21 \), \( \tan(20^{\circ})\approx0.364 \), \( NM=\frac{21}{0.364}\approx57.7 \), but the options given in the dropdown (from the image, the dropdown has 20, 70, 90? Wait, maybe the options are for the angle of \( \angle L \) (70), the ratio is tangent, and \( NM \approx57.7 \)? But the options in the dropdown for the last part are 20, 70, 90? Wait, maybe there is a mis - reading. Wait, maybe \( LN = 21 \), and we use \( \cot M=\frac{NM}{LN} \), \( \cot(20^{\circ})=\frac{1}{\tan(20^{\circ})}\approx2.747 \), so \( NM = 21\times2.747\approx57.7 \), but the options given are 20, 70, 90? Wait, maybe the problem has a typo, but according to the steps:

Answer:

The measure of angle L is \( \boldsymbol{70} \)°.
The trigonometric ratio that uses \( \angle M \) and \( LN \) to solve for \( NM \) is \( \boldsymbol{\text{tangent}} \).
The length of \( NM \), to the nearest tenth, is approximately \( \boldsymbol{57.7} \) (but if we consider the given options 20, 70, 90, maybe there is an error in the problem or the options. However, based on calculation, if we follow the steps, the angle L is \( 70^{\circ} \), ratio is tangent, and \( NM\approx57.7 \), but if we have to choose from 20, 70, 90, maybe a mistake. But according to the angle calculation, angle L is \( 70^{\circ} \).