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I can't verify that it's a right angle triangle

The triangle in this image is not a right-angled triangle, but we can still find the angles \( x \), \( y \), and \( z \) using the **Cosine Rule** and **Sine Rule**.

### Step-by-Step Solution

1. **Using the Cosine Rule** to find one of the angles, say \( z \).
   - The **Cosine Rule** formula is:
     \[
     \cos(z) = \frac{a^2 + b^2 - c^2}{2ab}
     \]
     where \( a = 10 \), \( b = 12 \), and \( c = 6 \).

   - Plugging in the values:
     \[
     \cos(z) = \frac{10^2 + 12^2 - 6^2}{2 \times 10 \times 12}
     \]
     \[
     \cos(z) = \frac{100 + 144 - 36}{240} = \frac{208}{240} = 0.8667
     \]
   - Solving for \( z \):
     \[
     z = \cos^{-1}(0.8667) \approx 30.0^\circ
     \]

2. **Using the Sine Rule** to find another angle, say \( x \).
   - The **Sine Rule** formula is:
     \[
     \frac{\sin(x)}{10} = \frac{\sin(z)}{6}
     \]
   - Since we have \( z = 30.0^\circ \), calculate \( \sin(30^\circ) = 0.5 \):
     \[
     \frac{\sin(x)}{10} = \frac{0.5}{6}
     \]
     \[
     \sin(x) = \frac{10 \times 0.5}{6} = \frac{5}{6} \approx 0.8333
     \]
   - Solving for \( x \):
     \[
     x = \sin^{-1}(0.8333) \approx 56.4^\circ
     \]

3. **Finding the final angle \( y \)**:
   - Since the sum of angles in a triangle is \( 180^\circ \), we have:
     \[
     y = 180^\circ - x - z
     \]
     \[
     y = 180^\circ - 56.4^\circ - 30.0^\circ
     \]
     \[
     y \approx 93.6^\circ
     \]

### Final Answer (to 1 decimal place):

- \( x \approx 56.4^\circ \)
- \( y \approx 93.6^\circ \)
- \( z \approx 30.0^\circ \)

Would you like further details or have any questions?

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Here are some related questions for further exploration:

1. How would the solution change if side lengths were different?
2. What if we used the Sine Rule first instead of the Cosine Rule?
3. Can the triangle inequality theorem confirm that this is a valid triangle?
4. How would you verify if the triangle is acute, obtuse, or right-angled?
5. What are some practical applications of the Cosine Rule in real-life problems?

**Tip:** When solving triangles, always start by identifying the known values and decide which trigonometric rule is most efficient based on the available information.

Find x, y, and z to 1 decimal place for each in the triangle with sides 6, 10, and 12.

Solve for angles x, y, and z in a triangle with side lengths 6, 10, and 12 using the Cosine Rule and Sine Rule. Includes a step-by-step solution suitable for students in Grades 10-12.

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