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.