Question

fall 2025 geometry ii wwva
right triangle relationships and trigonometry
which measures are accurate regarding triangle jkl? choose two correct answers.

Explanation:

Step1: Find the measure of angle \(K\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(m\angle K=180^{\circ}-(58^{\circ} + 38^{\circ})\)

$$m\angle K=180^{\circ}-96^{\circ}=84^{\circ}$$

Step2: Use the Law of Sines to find \(KL\)

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = KL\), \(A = 58^{\circ}\), \(b=2.3\), \(B = 38^{\circ}\)
\(\frac{KL}{\sin58^{\circ}}=\frac{2.3}{\sin38^{\circ}}\)

$$KL=\frac{2.3\times\sin58^{\circ}}{\sin38^{\circ}}$$

$$KL=\frac{2.3\times0.848}{0.616}\approx3.2$$

Step3: Use the Law of Sines to find \(k\)

Let \(a = k\), \(A = 84^{\circ}\), \(b = 2.3\), \(B=38^{\circ}\)
\(\frac{k}{\sin84^{\circ}}=\frac{2.3}{\sin38^{\circ}}\)

$$k=\frac{2.3\times\sin84^{\circ}}{\sin38^{\circ}}$$

$$k=\frac{2.3\times0.994}{0.616}\approx3.7$$

Answer:

\(KL\approx3.2\) units, \(k\approx3.7\) units