Question

use the diagram to complete the statements. the measure of angle l is . the trigonometric ratio that uses ∠m and ln to solve for nm is . the length of nm is approximately to the nearest tenth.

Explanation:

Step1: Identify the trigonometric ratio for finding NM

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In right - triangle $LMN$, $\angle M = 20^{\circ}$, the opposite side to $\angle M$ is $LN = 21$ and the adjacent side is $NM$. So the trigonometric ratio is $\tan M=\frac{LN}{NM}$.

Step2: Solve for NM

We have $\tan(20^{\circ})=\frac{21}{NM}$, then $NM=\frac{21}{\tan(20^{\circ})}$. Since $\tan(20^{\circ})\approx0.364$, $NM=\frac{21}{0.364}\approx57.7$.

Step3: Find the measure of angle L

In a right - triangle, the sum of the two non - right angles is $90^{\circ}$. So $\angle L+\angle M = 90^{\circ}$. Then $\angle L=90^{\circ}-\angle M$. Substituting $\angle M = 20^{\circ}$, we get $\angle L = 70^{\circ}$.

Answer:

The trigonometric ratio that uses $\angle M$ and $LN$ to solve for $NM$ is $\tan M=\frac{LN}{NM}$. The length of $NM$, to the nearest tenth, is $57.7$. The measure of angle $L$ is $70^{\circ}$.