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We are given a triangle \( \triangle ABC \) where \( BM \) is a median, meaning it divides the opposite side \( AC \) into two equal segments, and it is also stated that \( BM = MC \). The angle at point \( A \) is labeled as \( 5x \) and the angle at point \( C \) is labeled as \( x \), and we are asked to find the value of angle \( x \).

### Steps to solve:

1. Since \( BM \) is the median and \( BM = MC \), the triangle is isosceles with \( \triangle BMC \) being equal in terms of side lengths.
   
2. In any triangle, the sum of the interior angles must equal 180°. Therefore, the sum of angles \( \angle A \), \( \angle B \), and \( \angle C \) in triangle \( \triangle ABC \) must satisfy:
   \[
   \angle A + \angle B + \angle C = 180^\circ
   \]

3. From the diagram, we know:
   - \( \angle A = 5x \)
   - \( \angle C = x \)
   
   So, let \( \angle B = y \). This gives us the equation:
   \[
   5x + y + x = 180^\circ
   \]
   Simplifying:
   \[
   6x + y = 180^\circ
   \]

4. Since \( \triangle BMC \) is isosceles, the angles at \( B \) and \( C \) opposite the equal sides \( BM \) and \( MC \) are equal. Thus, \( \angle B = \angle C = x \).

5. Substituting \( y = x \) into the equation \( 6x + y = 180^\circ \):
   \[
   6x + x = 180^\circ
   \]
   \[
   7x = 180^\circ
   \]
   Solving for \( x \):
   \[
   x = \frac{180^\circ}{7} \approx 25.71^\circ
   \]

Thus, the measure of angle \( x \) is approximately \( 25.71^\circ \).

Would you like a detailed explanation of any part of the solution?

### Related Questions:
1. What are the properties of an isosceles triangle?
2. How do we apply the sum of angles in a triangle to solve for unknown angles?
3. How would the solution change if \( BM \) was not equal to \( MC \)?
4. What other properties can be deduced from a median in a triangle?
5. How would this problem change if the triangle was scalene instead of isosceles?

**Tip:** Always remember that the sum of the interior angles of any triangle is \( 180^\circ \), which is crucial for solving most angle-related problems.

In the given figure, BM is a median and BM = MC. What is the value of angle x?

Learn how to find the value of angle x in a triangle where BM is a median and BM = MC. This problem uses key geometric concepts such as the sum of interior angles and the properties of isosceles triangles. Suitable for students in Grades 9-10.

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