Question

∠1 and angle, 2∠2 are complementary angles. If m, angle, 1, equals, left parenthesis, x, plus, 2, right parenthesis, degreesm∠1=(x+2)
∘
and m, angle, 2, equals, left parenthesis, x, minus, 26, right parenthesis, degreesm∠2=(x−26)
∘
, then find the measure of angle, 1∠1.

Answer 1

Two angles are complementary if the sum of their measures is \(90^\circ\).

Let:
- \( m\angle 1 = (x + 2)^\circ \)
- \( m\angle 2 = (x - 26)^\circ \)

According to the definition of complementary angles:
\[
m\angle 1 + m\angle 2 = 90^\circ
\]

Substituting the expressions for \(m\angle 1\) and \(m\angle 2\):
\[
(x + 2) + (x - 26) = 90
\]

Combining like terms:
\[
2x - 24 = 90
\]

Now, solve for \(x\):
\[
2x = 90 + 24
\]
\[
2x = 114
\]
\[
x = 57
\]

Now that we have \(x\), we can find the measure of \(m\angle 1\):
\[
m\angle 1 = x + 2 = 57 + 2 = 59^\circ
\]

Thus, the measure of angle \(1\) is:
\[
\boxed{59}
\]