Question

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 angle L

In right - triangle LMN, since it's a right - triangle and one angle is 20° and another is 90°, we use the angle - sum property of a triangle (sum of angles in a triangle is 180°). Let \(\angle L=x\), \(\angle M = 20^{\circ}\), \(\angle N=90^{\circ}\). Then \(x + 20^{\circ}+90^{\circ}=180^{\circ}\), so \(x=180^{\circ}-(90^{\circ} + 20^{\circ})=70^{\circ}\).

Step2: Identify the trigonometric ratio for NM

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 the trigonometric ratio is \(\tan M=\frac{LN}{NM}\).

Step3: Calculate the length of NM

We know that \(\tan M=\frac{LN}{NM}\), given \(M = 20^{\circ}\) and \(LN = 21\). Then \(\tan20^{\circ}=\frac{21}{NM}\), and \(NM=\frac{21}{\tan20^{\circ}}\). Since \(\tan20^{\circ}\approx0.364\), \(NM=\frac{21}{0.364}\approx57.7\).

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 \(57.7\).