To solve this problem, we will use trigonometric relations involving the Law of Cosines and the Law of Sines. Follow these steps:
1. Calculate YZ using the Law of Cosines:
In triangle XYZ, we know:
- m∠ZY = 75°
- XY = 10
- XZ = 13
The Law of Cosines states:
\(\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]\)
Here, \(\( c = YZ \)\), \(\( a = XY = 10 \)\), \(\( b = XZ = 13 \)\), and \(\( C = m∠ZY = 75° \)\).
So,
\(\[ YZ^2 = XY^2 + XZ^2 - 2 \cdot XY \cdot XZ \cdot \cos(m∠ZY) \]\)
\(\[ YZ^2 = 10^2 + 13^2 - 2 \cdot 10 \cdot 13 \cdot \cos(75°) \]\)
First, calculate the cosine of 75°:
\(\[ \cos(75°) \approx 0.2588 \]\)
Now plug in these values:
\(\[ YZ^2 = 100 + 169 - 2 \cdot 10 \cdot 13 \cdot 0.2588 \]\)
\(\[ YZ^2 = 269 - 67.328 \]\)
\(\[ YZ^2 = 201.672 \]\)
Taking the square root to get YZ:
\(\[ YZ = \sqrt{201.672} \approx 14.2 \]\)
2. Calculate the measure of angle ZZ using the Law of Sines:
Next, we use the Law of Sines, which states:
\(\[ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \]\)
Let's denote angle X as ∠ZX (since this is the corner we're actually solving for).
\(\[ \frac{\sin(X)}{XZ} = \frac{\sin(75°)}{YZ} \]\)
Plug in known values:
\(\[ \frac{\sin(X)}{13} = \frac{\sin(75°)}{14.2} \]\)
Calculate sine of 75°:
\(\[ \sin(75°) \approx 0.9659 \]\)
Now,
\(\[ \frac{\sin(X)}{13} = \frac{0.9659}{14.2} \]\)
\(\[ \sin(X) = 13 \cdot \frac{0.9659}{14.2} \]\)
\(\[ \sin(X) \approx \frac{12.5567}{14.2} \]\)
\(\[ \sin(X) \approx 0.8843 \]\)
To get angle ∠X:
\(\[ X = \arcsin(0.8843) \]\)
\(\[ X \approx 62.3° \]\)
Therefore, the measure of angle ∠ZX (or mZZ, as denoted) is approximately:
\(\[ mZZ \approx 62.3° \]\)
So, the measure of mZZ to the nearest tenth is:
\(\[ \boxed{62.3°} \]\)
