∠1 and angle, 2∠2 are complementary angles. If m, angle, 1, equals, left parenthesis, 2, x, plus, 27, right parenthesis, degreesm∠1=(2*x*+27)∘ and m, angle, 2, equals, left parenthesis, 2, x, minus, 21, right parenthesis, degreesm∠2=(2*x*−21)∘, then find the measure of angle, 1∠1.

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.

Learn how to find the measure of angle 1 when given that angles 1 and 2 are complementary. This problem involves solving a linear equation derived from the definition of complementary angles. Suitable for Grades 7-9 students.

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