Question

solving for measures of right triangles
use the diagram to complete the statements.
the measure of angle l is \boxed{\checkmark}^\circ.
the trigonometric ratio that uses $\angle m$ and ln to solve for nm is \boxed{\checkmark}.
the length of nm, to the nearest tenth, is approximately \boxed{\checkmark}.

Explanation:

Step1: Calculate angle L

In right triangle LNM, the sum of angles is $180^\circ$. We know $\angle N=90^\circ$, $\angle M=20^\circ$.
$\angle L = 180^\circ - 90^\circ - 20^\circ = 70^\circ$

Step2: Identify trigonometric ratio

For $\angle M$, side $LN$ is opposite, $NM$ is adjacent. The ratio of opposite to adjacent is tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, so we use tangent to relate $\angle M$, $LN$, and $NM$.

Step3: Calculate length of NM

We know $\tan(20^\circ)=\frac{LN}{NM}$, rearrange to solve for $NM$:
$NM = \frac{LN}{\tan(20^\circ)}$
Substitute $LN=21$, $\tan(20^\circ)\approx0.3640$:
$NM \approx \frac{21}{0.3640} \approx 57.7$

Answer:

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