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To calculate the angles \( z \) and \( y \) in the given triangle, we can use trigonometric ratios. Here’s the process:

### Step 1: Identify the sides
- The side opposite angle \( z \) (which is labeled with the blue angle marker) is 12 cm.
- The side adjacent to angle \( z \) is 14.5 cm.

### Step 2: Calculate angle \( z \)
We'll use the tangent function, defined as:

\[
\tan(z) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{14.5}
\]

Let's calculate this:

\[
\tan(z) = \frac{12}{14.5} \approx 0.8276
\]

Now, find \( z \) by taking the inverse tangent (arctan) of 0.8276:

\[
z = \arctan(0.8276) \approx 39.3^\circ
\]

### Step 3: Calculate angle \( y \)
We know that the sum of all angles in a triangle is 180 degrees. In this right triangle:

\[
y = 90^\circ - z = 90^\circ - 39.3^\circ \approx 50.7^\circ
\]

### Final Answers:
- The size of angle \( z \) is approximately **39.3°** (to 1 decimal place).
- The size of angle \( y \) is approximately **50.7°** (to 1 decimal place).

Would you like more details, or do you have any other questions?

Here are 5 related questions:

1. How would you find angle \( z \) using the sine function?
2. Can you calculate the hypotenuse of the triangle using the sides given?
3. What is the relationship between angles \( y \) and \( z \) in this triangle?
4. How would changing the length of the opposite side affect angle \( z \)?
5. What trigonometric function would you use if you knew the hypotenuse and the opposite side for angle \( z \)?

**Tip:** When using trigonometric functions, make sure your calculator is set to the correct mode (degrees or radians) based on the question.

Learn how to calculate angles z and y in a right triangle using trigonometric ratios. Step-by-step solution provided with detailed calculations. Suitable for high school students studying trigonometry.

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