To solve for the measure of [tex]\(\angle z\)[/tex], we will employ the law of cosines. The law of cosines is stated as:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \][/tex]
Here, [tex]\( c = 16 \)[/tex], [tex]\( a = 18 \)[/tex], and [tex]\( b = 19 \)[/tex]. We want to solve for the angle [tex]\( C \)[/tex], which is [tex]\(\angle z\)[/tex].
1. Insert the known values into the equation:
[tex]\[ 16^2 = 18^2 + 19^2 - 2 \cdot 18 \cdot 19 \cdot \cos(C) \][/tex]
2. Calculate the squares:
[tex]\[ 256 = 324 + 361 - 2 \cdot 18 \cdot 19 \cdot \cos(C) \][/tex]
3. Simplify the equation:
[tex]\[ 256 = 685 - 684 \cdot \cos(C) \][/tex]
4. Isolate the term with [tex]\(\cos(C)\)[/tex]:
[tex]\[ 256 - 685 = -684 \cdot \cos(C) \][/tex]
[tex]\[ -429 = -684 \cdot \cos(C) \][/tex]
5. Divide both sides by [tex]\(-684\)[/tex]:
[tex]\[ \cos(C) = \frac{-429}{-684} \][/tex]
[tex]\[ \cos(C) = \frac{429}{684} \][/tex]
6. Simplify the fraction:
[tex]\[ \cos(C) = 0.6271929824561403 \][/tex]
7. Use the inverse cosine function to find angle [tex]\(C\)[/tex]:
[tex]\[ C = \cos^{-1}(0.6271929824561403) \][/tex]
The result in radians is:
[tex]\[ C \approx 0.8928523578433809 \text{ radians} \][/tex]
8. Convert the angle from radians to degrees:
[tex]\[ C \approx 51.156671832730034 \text{ degrees} \][/tex]
9. Round to the nearest whole degree:
[tex]\[ \angle z \approx 51^\circ \][/tex]
Therefore, the measure of [tex]\(\angle z\)[/tex], to the nearest whole degree, is [tex]\(51^\circ\)[/tex]. The correct answer is:
[tex]\[ \boxed{51^\circ} \][/tex]
