Answer:
C. [tex]Z=51^{\circ}[/tex]
Step-by-step explanation:
We have been given a triangle. We are asked to find the measure of angle Z using Law of cosines.
Law of cosines: [tex]c^2=a^2+b^2-2ab\cdot \text{cos}(C)[/tex], where, a, b and c are sides opposite to angles A, B and C respectively.
Upon substituting our given values in law of cosines, we will get:
[tex]16^2=19^2+18^2-2(19)(18)\cdot \text{cos}(Z)[/tex]
[tex]256=361+324-684\cdot \text{cos}(Z)[/tex]
[tex]256=685-684\cdot \text{cos}(Z)[/tex]
[tex]256-685=685-685-684\cdot \text{cos}(Z)[/tex]
[tex]-429=-684\cdot \text{cos}(Z)[/tex]
[tex]\frac{-429}{-684}=\frac{-684\cdot \text{cos}(Z)}{-684}[/tex]
[tex]0.627192982456=\text{cos}(Z)[/tex]
[tex]\text{cos}(Z)=0.627192982456[/tex]
Now, we will use inverse cosine or arc-cos to solve for angle Z as:
[tex]Z=\text{cos}^{-1}(0.627192982456)[/tex]
[tex]Z=51.1566718^{\circ}[/tex]
[tex]Z\approx 51^{\circ}[/tex]
Therefore, the measure of angle Z is approximately 51 degrees.