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.