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The given problem shows a triangle where the angles are labeled as \( x \), \( y \), and \( z \), and we are provided the ratio \( x : y : z = 2 : 3 : 4 \). Since this is a triangle, the sum of all interior angles must be \( 180^\circ \).

To solve this, let's use the following steps:

1. The ratio of the angles is given as \( x : y : z = 2 : 3 : 4 \).
   - Let \( x = 2k \), \( y = 3k \), and \( z = 4k \) where \( k \) is a constant multiplier.

2. The sum of the angles in any triangle is \( 180^\circ \), so:
   \[
   x + y + z = 180^\circ
   \]
   Substituting the expressions for \( x \), \( y \), and \( z \):
   \[
   2k + 3k + 4k = 180^\circ
   \]

3. Combine the terms:
   \[
   9k = 180^\circ
   \]

4. Solve for \( k \):
   \[
   k = \frac{180^\circ}{9} = 20^\circ
   \]

5. Now, substitute \( k = 20^\circ \) back into the expressions for \( x \), \( y \), and \( z \):
   - \( x = 2k = 2 \times 20^\circ = 40^\circ \)
   - \( y = 3k = 3 \times 20^\circ = 60^\circ \)
   - \( z = 4k = 4 \times 20^\circ = 80^\circ \)

Thus, the value of \( x \) is \( 40^\circ \).

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

Here are 5 related questions to further expand your understanding:

1. How would the solution change if the ratio \( x : y : z \) were different?
2. What is the general method to solve for angles in a triangle when given a ratio of angles?
3. Why must the sum of the angles in any triangle always be \( 180^\circ \)?
4. Can you use the same method to solve a quadrilateral where the sum of angles is \( 360^\circ \)?
5. How would you solve for the sides of a triangle if given a ratio similar to the angles?

**Tip**: When working with angle ratios, always remember that the sum of the angles in a triangle is fixed at \( 180^\circ \), which gives you a direct way to find the value of each angle.

If x : y : z = 2 : 3 : 4, find the value of x.

Learn how to solve for the value of x in a triangle where the ratio of the angles is x:y:z = 2:3:4. This step-by-step guide will walk you through using the angle sum theorem to find the missing angles. Suitable for students in Grades 8-10.

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