To solve this problem, we use the fact that complementary angles add up to $90^\circ$. The given angles are:

• $m\angle1 = (x + 24)^\circ$
• $m\angle2 = (5x - 18)^\circ$

Since these two angles are complementary, their sum equals $90^\circ$:

$(x + 24) + (5x - 18) = 90$

Now, solve for $x$:

1. Combine like terms: $x + 5x + 24 - 18 = 90$ $6x + 6 = 90$

2. Subtract 6 from both sides: $6x = 84$

3. Divide both sides by 6: $x = 14$

Now that we have $x = 14$, we can find the measure of $\angle1$:

$m\angle1 = x + 24 = 14 + 24 = 38^\circ$

Thus, the measure of $\angle1$ is $38^\circ$.

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Would you like a further breakdown of the steps? Here are some related questions to deepen your understanding:

1. How do you determine if two angles are complementary?
2. Can you find the measure of $\angle2$ using the value of $x$?
3. What is the difference between complementary and supplementary angles?
4. How would the equation change if the angles were supplementary instead of complementary?
5. What happens to $\angle1$ and $\angle2$ if $x$ becomes negative?

Tip: Always double-check the conditions of a problem (like complementary or supplementary) to make sure you use the right relationships!