Question

solving for side lengths of right triangles
solving for measures of right triangles
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, to the nearest tenth, is approximately

Explanation:

Step1: Find measure of angle L

In right - triangle LNM, angle M = 20°, and since the sum of angles in a triangle is 180° and one angle is 90°, angle L=180° - 90° - 20° = 70°.

Step2: Identify trigonometric ratio

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. To solve for NM using $\angle M$ and LN, the trigonometric ratio is $\tan$. Because $\tan M=\frac{LN}{NM}$, so $NM = \frac{LN}{\tan M}$.

Step3: Calculate length of NM

Given LN = 21 and M = 20°. Then $NM=\frac{21}{\tan(20^{\circ})}$. Since $\tan(20^{\circ})\approx0.364$, $NM=\frac{21}{0.364}\approx57.7$.

Answer:

The measure of angle L is 70°.
The trigonometric ratio that uses $\angle M$ and LN to solve for NM is $\tan$.
The length of NM, to the nearest tenth, is approximately 57.7.