https://file.aiquickdraw.com/mathbot/file/1726684781548hguxzq2q.jpg

The image depicts a right-angled triangle \( \triangle ABC \), where we are asked to find the angle \( z \) at point \( A \). The right angle is at \( C \), and the side lengths \( AB \) (base) and \( AC \) (height) can be approximated using the grid.

### Steps to solve:
1. **Identify the sides**:  
   - Side \( AB \) (base) appears to be 10 units long.  
   - Side \( AC \) (height) appears to be 6 units long.  

2. **Use trigonometry**:  
   Since \( z \) is the angle at \( A \), and we have the opposite and adjacent sides relative to \( z \), we can use the tangent function:
   \[
   \tan(z) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{10} = 0.6
   \]
   
3. **Find the angle**:
   \[
   z = \tan^{-1}(0.6)
   \]
   Let's calculate this value to 1 decimal place.

### Calculation

I'll calculate this now.The value of \( z \), correct to 1 decimal place, is \( 31.0^\circ \).

Do you need further details or have any questions? Here are 5 related questions to explore:

1. What is the general process for using trigonometric ratios to find an unknown angle?
2. How can we determine if a triangle is right-angled just by looking at its sides?
3. What are the different trigonometric functions used for solving triangles?
4. How would you find the length of a missing side if two sides and one angle are known?
5. Can the Pythagorean theorem be applied to this triangle, and if so, how?

**Tip:** When dealing with right-angled triangles, always identify which sides are adjacent, opposite, or hypotenuse relative to the angle you're working with!

Work out the value of z in a right-angled triangle, given the side lengths and grid for approximation.

Learn how to calculate the angle z in a right-angled triangle using trigonometric ratios. The problem involves determining side lengths using a grid and applying the tangent formula to find the angle. Suitable for Grades 8-10 students studying trigonometry.

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