Complementary angles are two angles whose measures add up to $90^\circ$. Given that $\angle 1$ and $\angle 2$ are complementary, we have the equation:

$m\angle 1 + m\angle 2 = 90^\circ$

Substituting the given expressions for $m\angle 1$ and $m\angle 2$:

$(2x + 27)^\circ + (2x - 21)^\circ = 90^\circ$

Now, let's solve this equation step by step:

1. Combine like terms:

$2x + 27 + 2x - 21 = 90$

This simplifies to:

$4x + 6 = 90$

2. Subtract 6 from both sides to isolate the term with $x$:

$4x = 84$

3. Divide both sides by 4:

$x = 21$

Now that we have $x = 21$, substitute it back into the expression for $m\angle 1$:

$m\angle 1 = 2x + 27 = 2(21) + 27 = 42 + 27 = 69^\circ$

So, the measure of $\angle 1$ is $69^\circ$.

Would you like a detailed explanation of any step? Here are some related questions to consider:

1. What would the measure of $\angle 2$ be with $x = 21$?
2. How would the equation change if the angles were supplementary instead of complementary?
3. What happens to the measures of the angles if $x$ is negative?
4. How would you solve the problem if the angles were expressed in terms of different variables?
5. What are some real-world applications of complementary angles?

Tip: Always verify your final answer by substituting it back into the original equations to ensure accuracy.