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.