Question

given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle’s sides relative to that acute angle. here are the approximate ratios for angle measures 55°, 65°, and 75°.

angle	55°	65°	75°
$\frac{\text{opposite leg length}}{\text{hypotenuse length}}$	0.82	0.91	0.97
$\frac{\text{opposite leg length}}{\text{adjacent leg length}}$	1.43	2.14	3.73

use the table to approximate $m\angle l$ in the triangle below.
triangle $jkl$ with right angle at $k$, $lk = 3.2$, $jl = 11.9$
choose 1 answer:
a $55°$
b $65°$
c $75°$

Explanation:

Step1: Identify sides relative to ∠L

In right triangle \( \triangle JKL \) with right angle at \( K \), for \( \angle L \):

• Adjacent leg: \( KL = 3.2 \)
• Hypotenuse: \( JL = 11.9 \)

Step2: Calculate the ratio \( \frac{\text{adjacent leg length}}{\text{hypotenuse length}} \)

\[
\frac{KL}{JL} = \frac{3.2}{11.9} \approx 0.269
\]

Step3: Compare with table values

The table gives the ratio \( \frac{\text{adjacent leg length}}{\text{hypotenuse length}} \) for \( 55^\circ \) as \( 0.57 \), for \( 65^\circ \) as \( 0.42 \), and for \( 75^\circ \) as \( 0.26 \). Our calculated ratio \( \approx 0.269 \) is closest to \( 0.26 \) (for \( 75^\circ \)).

Answer:

C \( 75^\circ \)