Certainly! Let's solve this step-by-step.
1. Define the unknown angle.
Let's denote the unknown angle as [tex]\( A \)[/tex].
2. Identify the supplement of the angle.
The supplement of an angle [tex]\( A \)[/tex] is [tex]\( 180^\circ - A \)[/tex].
3. Identify the complement of the angle.
The complement of an angle [tex]\( A \)[/tex] is [tex]\( 90^\circ - A \)[/tex].
4. Identify the supplement of the complement of the angle.
The supplement of [tex]\( 90^\circ - A \)[/tex] is [tex]\( 180^\circ - (90^\circ - A) \)[/tex].
Simplifying, we get:
[tex]\[
180^\circ - 90^\circ + A = 90^\circ + A
\][/tex]
5. Set up the relationship given in the problem.
According to the problem, the supplement of the angle ([tex]\( 180^\circ - A \)[/tex]) is 36° less than twice the supplement of the complement of the angle ([tex]\( 2 \times (90^\circ + A) - 36^\circ \)[/tex]).
Therefore, we can write the equation:
[tex]\[
180^\circ - A = 2 \times (90^\circ + A) - 36^\circ
\][/tex]
6. Simplify the equation.
Simplify the right-hand side of the equation:
[tex]\[
2 \times (90^\circ + A) - 36^\circ = 180^\circ + 2A - 36^\circ
\][/tex]
Which reduces to:
[tex]\[
144^\circ + 2A
\][/tex]
So the equation becomes:
[tex]\[
180^\circ - A = 144^\circ + 2A
\][/tex]
7. Solve for [tex]\( A \)[/tex].
To solve for [tex]\( A \)[/tex], move all terms involving [tex]\( A \)[/tex] to one side of the equation:
[tex]\[
180^\circ - 144^\circ = 2A + A
\][/tex]
Simplify:
[tex]\[
36^\circ = 3A
\][/tex]
Divide by 3:
[tex]\[
A = 12^\circ
\][/tex]
8. Calculate the supplement of the angle.
The supplement of the angle is:
[tex]\[
180^\circ - A = 180^\circ - 12^\circ = 168^\circ
\][/tex]
So, the measure of the supplement of the angle is [tex]\( 168^\circ \)[/tex].
