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To find the measure of angle \( \angle XYZ \) in the given right triangle, we can use trigonometric functions. Here's the step-by-step process:

### Step 1: Identify the Triangle Sides
- **Hypotenuse**: \( YZ = 8\sqrt{3} \)
- **Adjacent Side to \( \angle XYZ \)**: \( XY = 4\sqrt{3} \)
- **Opposite Side to \( \angle XYZ \)**: Since this is a right triangle, \( \triangle XYZ \) has a right angle at \( X \), making the opposite side of \( \angle XYZ \) the side that is perpendicular to \( XY \), which is \( 8\sqrt{3} \) because \( Z \) is the hypotenuse opposite the right angle.

### Step 2: Select the Appropriate Trigonometric Function
Since we have the adjacent side (\( XY \)) and the hypotenuse (\( YZ \)), we use the cosine function:
\[
\cos(\angle XYZ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{XY}{YZ} = \frac{4\sqrt{3}}{8\sqrt{3}}.
\]

### Step 3: Simplify the Ratio
\[
\cos(\angle XYZ) = \frac{4\sqrt{3}}{8\sqrt{3}} = \frac{4}{8} = \frac{1}{2}.
\]

### Step 4: Find the Angle
The angle whose cosine is \( \frac{1}{2} \) is \( 60^\circ \):
\[
\angle XYZ = 60^\circ.
\]

So, the measure of \( \angle XYZ \) is **60 degrees**.

### Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would you find \( \angle XYZ \) using the sine or tangent function?
2. What is the length of the side opposite \( \angle XYZ \) in this triangle?
3. Can you find the measure of the other non-right angle in this triangle?
4. How can you verify that the sum of angles in a triangle is \( 180^\circ \)?
5. What is the relationship between the sides of a 30-60-90 triangle?

**Tip:** In a right triangle, the cosine of an angle is equal to the ratio of the length of the adjacent side to the hypotenuse.

What is the measure in degrees of ∠XYZ in the figure shown?

Learn how to find the measure of angle ∠XYZ in a right triangle using trigonometric functions. This problem covers cosine function application and involves identifying the adjacent side and hypotenuse. Suitable for high school students in Grades 9-12.

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