To solve this problem and find the size of angle $\theta$, we can use the trigonometric ratio known as the tangent function, which is defined as:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

In the given right triangle:

• The length of the side opposite $\theta$ is 9 cm.
• The length of the side adjacent to $\theta$ is 13 cm.

Thus, we can compute:

$\tan \theta = \frac{9}{13}$

Next, to find $\theta$, we take the inverse tangent (also called arctan) of the ratio:

$\theta = \tan^{-1} \left( \frac{9}{13} \right)$

Using a calculator to find $\theta$:

$\theta = \tan^{-1}(0.6923) \approx 34.85^\circ$

Rounding to the nearest integer, the size of angle $\theta$ is approximately $35^\circ$.

Would you like more details or have any other questions?

Here are 5 related questions you might want to explore:

1. What are the other trigonometric ratios for this triangle (sine and cosine)?
2. How do you calculate the hypotenuse using the Pythagorean theorem?
3. How does rounding affect trigonometric calculations in practice?
4. How would the problem change if the triangle were not a right triangle?
5. Can you solve for all angles in the triangle using only the side lengths?

Tip: Always make sure your calculator is set to the correct mode (degrees or radians) based on the problem's requirements!