To solve this problem step by step:
[tex]$\triangle JKL$[/tex] has [tex]\( j = 7 \)[/tex], [tex]\( k = 11 \)[/tex], and [tex]\( m \angle J = 18^\circ \)[/tex]. We need to determine the possible measures of angle [tex]\( K \)[/tex].
1. Identify the ambiguous case:
Triangle JKL meets the SSA (Side-Side-Angle) criteria, which means it is the ambiguous case.
2. Substitute the known values into the Law of Sines:
[tex]\[
\frac{\sin(18^\circ)}{7} = \frac{\sin(K)}{11}
\][/tex]
3. Cross multiply:
[tex]\[
11 \sin(18^\circ) = 7 \sin(K)
\][/tex]
4. Substitute [tex]\( \sin(18^\circ) \approx 0.309 \)[/tex]:
[tex]\[
11 \times 0.309 \approx 3.399
\][/tex]
5. Solve for [tex]\( \sin(K) \)[/tex]:
[tex]\[
7 \sin(K) \approx 3.399
\][/tex]
[tex]\[
\sin(K) \approx \frac{3.399}{7} \approx 0.486
\][/tex]
6. Find [tex]\( K \)[/tex] using the inverse sine function:
[tex]\[
K \approx \arcsin(0.486)
\][/tex]
Using a calculator:
[tex]\[
K \approx 29^\circ
\][/tex]
However, because this is the ambiguous case, we also need to consider the supplementary angle, which is:
[tex]\[
180^\circ - 29^\circ = 151^\circ
\][/tex]
Therefore, the possible measures of angle [tex]\( K \)[/tex], rounded to the nearest degree, are 29° and 151° respectively.
7. Complete the given statements:
Triangle JKL meets the SSA (Side-Side-Angle) criteria, which means it is the ambiguous case.
Substitute the known values into the law of sines:
[tex]\[
\frac{\sin(18^\circ)}{7} = \frac{\sin(K)}{11}
\][/tex]
Cross multiply:
[tex]\[
11 \sin(18^\circ) = 3.399
\][/tex]
Solve for the measure of angle [tex]\( K \)[/tex], and use a calculator to determine the value.
Round to the nearest degree:
[tex]\[
m \angle K \approx 29^\circ
\][/tex]
However, because this is the ambiguous case, the measure of angle [tex]\( K \)[/tex] could also be:
[tex]\[
151^\circ
\][/tex]
