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To find the size of angle \(\angle XYZ\) in the right-angled triangle, we can use trigonometry. Specifically, we will use the tangent function, since we have the lengths of the opposite and adjacent sides relative to angle \(\angle XYZ\).

### Step-by-step:

1. We are given:
   - The length of the opposite side (XY) = 4 cm
   - The length of the adjacent side (XZ) = 13 cm

2. The tangent of an angle in a right-angled triangle is given by:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
   \]
   Substituting the values for \(\angle XYZ\):
   \[
   \tan(\angle XYZ) = \frac{4}{13}
   \]

3. To find \(\angle XYZ\), take the arctangent (inverse tangent) of \(\frac{4}{13}\):
   \[
   \angle XYZ = \tan^{-1}\left(\frac{4}{13}\right)
   \]

Let me calculate that.The size of angle \(\angle XYZ\) is \(17.1^\circ\), rounded to one decimal place.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would you calculate the other angles in the triangle?
2. What trigonometric function could be used if you were given the hypotenuse instead of the adjacent side?
3. Can you find the length of the hypotenuse using Pythagoras' theorem?
4. How can the sine function be used to calculate \(\angle XYZ\)?
5. What would the value of \(\angle XYZ\) be if the opposite side were 5 cm instead of 4 cm?

**Tip:** Always ensure your calculator is set to degrees, not radians, when working with angles in degrees.

Find the size of angle XYZ. Triangle XYZ with side XY = 4 cm, XZ = 13 cm, right angle at X.

Learn how to calculate the angle XYZ in a right-angled triangle where side XY is 4 cm, side XZ is 13 cm, and the right angle is at X. The solution uses trigonometric principles such as the tangent function to find the angle.

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