Question

∠1 and angle, 2∠2 are complementary angles. If m, angle, 1, equals, left parenthesis, 4, x, minus, 13, right parenthesis, degreesm∠1=(4x−13)
∘
and m, angle, 2, equals, left parenthesis, x, minus, 12, right parenthesis, degreesm∠2=(x−12)
∘
, then find the measure of angle, 2∠2.

Answer 1

Complementary angles are two angles whose measures add up to 90 degrees. Given that \( m\angle 1 = 4x - 13 \) and \( m\angle 2 = x - 12 \), we can set up the equation:

\[
m\angle 1 + m\angle 2 = 90
\]

Substituting the expressions for \( m\angle 1 \) and \( m\angle 2 \):

\[
(4x - 13) + (x - 12) = 90
\]

Now, combine like terms:

\[
4x + x - 13 - 12 = 90
\]
\[
5x - 25 = 90
\]

Next, add 25 to both sides:

\[
5x = 115
\]

Now, divide by 5:

\[
x = 23
\]

Now that we have the value of \( x \), we can find \( m\angle 2 \):

\[
m\angle 2 = x - 12 = 23 - 12 = 11
\]

Thus, the measure of angle \( 2 \) is:

\[
\boxed{11}
\]

To verify, we can find \( m\angle 1 \):

\[
m\angle 1 = 4x - 13 = 4(23) - 13 = 92 - 13 = 79
\]

Now we check if they are complementary:

\[
m\angle 1 + m\angle 2 = 79 + 11 = 90
\]

Indeed, the angles are complementary. Therefore, the final answer is:

\[
\boxed{11}
\]