To find the size of angle $\angle XZY$ in triangle XYZ, we can use the Law of Sines or the Law of Cosines. However, because we are given two sides and the included angle, the Law of Cosines is more appropriate for solving this problem.

Step 1: Apply the Law of Cosines

The Law of Cosines states that for any triangle $ABC$: $c^2 = a^2 + b^2 - 2ab \cos C$ where $c$ is the side opposite angle $C$, and $a$ and $b$ are the other two sides.

In triangle XYZ:

• $XY = c = 15 \, \text{cm}$
• $YZ = a = 10 \, \text{cm}$
• $XZ = b$
• $\angle YXZ = C = 31^\circ$

We first need to find the length of $XZ$ using the Law of Cosines.

Step 2: Calculate $XZ$

$XZ^2 = XY^2 + YZ^2 - 2 \times XY \times YZ \times \cos(\angle YXZ)$ $XZ^2 = 15^2 + 10^2 - 2 \times 15 \times 10 \times \cos(31^\circ)$ $XZ^2 = 225 + 100 - 300 \times \cos(31^\circ)$ $XZ^2 = 325 - 300 \times 0.8572$ $XZ^2 = 325 - 257.16$ $XZ^2 = 67.84$ $XZ = \sqrt{67.84} \approx 8.24 \, \text{cm}$

Step 3: Apply the Law of Sines to find $\angle XZY$

The Law of Sines states: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ We use the ratios involving $\angle XZY$ and $\angle YXZ$: $\frac{\sin(\angle XZY)}{YZ} = \frac{\sin(\angle YXZ)}{XZ}$ $\frac{\sin(\angle XZY)}{10} = \frac{\sin(31^\circ)}{8.24}$ $\sin(\angle XZY) = \frac{10 \times \sin(31^\circ)}{8.24}$ $\sin(\angle XZY) = \frac{10 \times 0.5150}{8.24}$ $\sin(\angle XZY) = \frac{5.15}{8.24} \approx 0.625$

Step 4: Find $\angle XZY$

$\angle XZY = \arcsin(0.625) \approx 38.7^\circ$

Thus, the size of $\angle XZY$ is approximately 38.7 degrees.

Would you like further details or have any questions about this solution? Here are five related questions you might explore:

1. How does the Law of Cosines differ from the Law of Sines in application?
2. What are the steps to determine if a triangle is obtuse?
3. How do you calculate the area of triangle XYZ using the given sides and angles?
4. Can you solve for the remaining angle $\angle YZX$ in triangle XYZ?
5. How does the cosine of an angle relate to the type of triangle (acute, obtuse, or right)?

Tip: When solving for angles in a triangle, always check if your calculated angles make sense within the context of the problem (e.g., whether they sum up to 180° in a triangle).