To find the measure of the larger of the two angles given that one angle is [tex]\(40^\circ\)[/tex] greater than 8 times its complement, let us proceed with the following steps:
1. Define the variables:
- Let [tex]\( x \)[/tex] be the measure of the given angle.
- The complement of [tex]\( x \)[/tex] will then be [tex]\( 90^\circ - x \)[/tex].
2. Set up the equation:
- According to the problem, one angle is [tex]\( 40^\circ \)[/tex] greater than 8 times its complement. This translates to the equation:
[tex]\[
x = 8(90 - x) + 40
\][/tex]
3. Solve the equation:
- Simplify inside the parentheses:
[tex]\[
x = 8 \cdot 90 - 8x + 40
\][/tex]
- Calculate:
[tex]\[
x = 720 - 8x + 40
\][/tex]
[tex]\[
x = 760 - 8x
\][/tex]
- Combine like terms by adding [tex]\( 8x \)[/tex] to both sides:
[tex]\[
x + 8x = 760
\][/tex]
[tex]\[
9x = 760
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{760}{9}
\][/tex]
4. Calculate the complement of [tex]\( x \)[/tex]:
- Using the formula for the complement, we have:
[tex]\[
\text{Complement of } x = 90^\circ - \frac{760}{9}
\][/tex]
- Simplify:
[tex]\[
\text{Complement of } x = \frac{810}{9} - \frac{760}{9} = \frac{50}{9}
\][/tex]
5. Determine the larger angle:
- Compare [tex]\( x \)[/tex] and its complement:
[tex]\[
x = \frac{760}{9}
\][/tex]
[tex]\[
\text{Complement of } x = \frac{50}{9}
\][/tex]
- Clearly, [tex]\(\frac{760}{9}\)[/tex] is larger than [tex]\(\frac{50}{9}\)[/tex].
6. Convert the larger angle to a decimal format if necessary:
- To get the final answer in degrees and rounded to two decimal places:
[tex]\[
\frac{760}{9} \approx 84.44^\circ
\][/tex]
Therefore, the measure of the larger angle is approximately [tex]\(84.44^\circ\)[/tex].
