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This problem is focused on finding the value of \( x \) given a ratio of the measures of the angles in a triangle.

The ratio of the angles is given as 3:4:5, and we need to solve for \( x \) in this extended ratio context. Here's the step-by-step breakdown from the provided solution:

1. **Set up the equation**: 
   - The ratio \( 3:4:5 \) can be translated into algebraic expressions for the angles. The angles are \( 3x \), \( 4x \), and \( 5x \).
   - According to the **Triangle Angle Sum Theorem**, the sum of the angles in any triangle is 180°. Therefore, you set up the equation:
   \[
   3x + 4x + 5x = 180
   \]
   
2. **Combine like terms**:
   - Combine all terms involving \( x \):
   \[
   12x = 180
   \]

3. **Solve for \( x \)**:
   - Divide both sides of the equation by 12 to isolate \( x \):
   \[
   x = \frac{180}{12} = 15
   \]

4. **Find the individual angles**:
   - Now that we have \( x = 15 \), substitute this value back into the expressions for the angles:
   \[
   \text{Smallest angle} = 3x = 3(15) = 45^\circ
   \]
   \[
   \text{Next angle} = 4x = 4(15) = 60^\circ
   \]
   \[
   \text{Largest angle} = 5x = 5(15) = 75^\circ
   \]

5. **Check**:
   - Ensure the sum of the angles equals 180°:
   \[
   45^\circ + 60^\circ + 75^\circ = 180^\circ
   \]
   The solution is correct.

Would you like more details on this problem, or have any specific questions?

### Related Questions:
1. How do you determine the sum of the angles in a polygon?
2. Can the triangle inequality theorem be applied to these angles?
3. What happens to the angles if the ratio changes to 2:3:4?
4. How would you solve a problem where one angle is known, and the ratio of the other two is given?
5. How does the concept of supplementary angles apply to non-triangle figures?

### Tip:
When working with angle ratios in triangles, always remember that the sum of all internal angles must be 180°.

If the ratio of the angles in a triangle is 3:4:5 and the smallest angle measures 36 degrees, what is x?

This problem involves solving for the variable x in a triangle where the angle ratio is given as 3:4:5. Using the Triangle Angle-Sum Theorem, we set up the equation 3x + 4x + 5x = 180 and solve for x. The smallest angle measures 36 degrees.

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