Sure, let's complete the statements step by step.
Given:
[tex]\( j = 7 \)[/tex]
[tex]\( k = 11 \)[/tex]
* [tex]\( \angle J = 18^\circ \)[/tex]
Step-by-Step Solution:
1. Identify the Ambiguous Case Criteria:
Triangle JKL meets the SSA (Side-Side-Angle) criteria, which means it is the ambiguous case. This situation occurs in a triangle where two sides and one non-included angle are known, and this can potentially yield two different triangles.
2. Law of Sines Setup:
Substitute the known values into the Law of Sines:
[tex]\[
\frac{\sin(18^\circ)}{7} = \frac{\sin(K)}{11}
\][/tex]
3. Cross Multiply to Isolate [tex]\(\sin(K)\)[/tex]:
To find [tex]\(\sin(K)\)[/tex], cross multiply:
[tex]\[
11 \sin(18^\circ) = 7 \sin(K)
\][/tex]
4. Solve for [tex]\(\sin(K)\)[/tex]:
Isolate [tex]\(\sin(K)\)[/tex]:
[tex]\[
\sin(K) = \frac{11 \sin(18^\circ)}{7}
\][/tex]
By calculating this value using a calculator:
[tex]\[
\sin(K) \approx 0.4855981340177745
\][/tex]
5. Determine Angle [tex]\( K \)[/tex]:
Use the inverse sine function (arcsin) to determine the measure of angle [tex]\( K \)[/tex]:
[tex]\[
K_1 = \sin^{-1}(0.4855981340177745)
\][/tex]
Using a calculator, we find:
[tex]\[
K_1 \approx 29.05^\circ \text{ (rounded to 29 degrees)}
\][/tex]
6. Ambiguous Case Consideration:
Because this is the ambiguous case (SSA condition), the measure of angle [tex]\( K \)[/tex] could also be:
[tex]\[
K_2 = 180^\circ - K_1
\][/tex]
Therefore:
[tex]\[
K_2 \approx 180^\circ - 29.05^\circ = 150.95^\circ \text{ (rounded to 151 degrees)}
\][/tex]
Final Statements:
1. 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 to get:
[tex]\[
11 \sin(18^\circ) = 7 \sin(K)
\][/tex]
4. Solve for the measure of angle [tex]\( K \)[/tex]:
[tex]\[
\sin(K) \approx 0.4855981340177745
\][/tex]
5. Use a calculator to determine the value:
[tex]\[
K_1 \approx 29^\circ
\][/tex]
6. However, because this is the ambiguous case, the measure of angle [tex]\( K \)[/tex] could also be:
[tex]\[
K_2 \approx 151^\circ
\][/tex]
So, the possible measures of angle [tex]\( K \)[/tex] are approximately [tex]\( 29^\circ \)[/tex] and [tex]\( 151^\circ \)[/tex].
