Since the sum of angle $M$ and angle $R$ is $90^\circ$, we can set up an equation based on the information given.

1. Let angle $M = (5x + 10)^\circ$.
2. Let angle $R = 55^\circ$.

Since the angles are complementary (sum to $90^\circ$), we have: $(5x + 10) + 55 = 90$

Now, we can solve for $x$:

1. Simplify the equation: $5x + 65 = 90$
2. Subtract $65$ from both sides: $5x = 25$
3. Divide by $5$ to isolate $x$: $x = 5$

Therefore, the value of $x$ is $5$.

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

Here are some related questions to consider:

1. What if the measure of angle $R$ were different? How would $x$ change?
2. What if the sum of angles $M$ and $R$ was a different angle measure?
3. How would you solve for $x$ if angle $M$ were given in terms of $x^2$?
4. Can you apply similar logic to find unknowns in triangles?
5. How would the equation look if $M$ and $R$ were supplementary instead of complementary?

Tip: Remember that complementary angles always add up to $90^\circ$, while supplementary angles add up to $180^\circ$.