Certainly! Let's solve this problem step-by-step.
1. Define the variables:
Let the measure of the angle be [tex]\( x \)[/tex].
2. Express the complement and supplement:
- The complement of the angle: [tex]\( 90^\circ - x \)[/tex]
- The supplement of the angle: [tex]\( 180^\circ - x \)[/tex]
3. Set up the equation:
According to the problem, six times the complement of the angle is 12 degrees less than twice its supplement. This relationship can be written as:
[tex]\[
6 \times (90^\circ - x) = 2 \times (180^\circ - x) - 12^\circ
\][/tex]
4. Distribute the constants:
Multiply out the terms inside the parentheses:
[tex]\[
540^\circ - 6x = 360^\circ - 2x - 12^\circ
\][/tex]
5. Combine like terms:
Simplify both sides of the equation:
[tex]\[
540^\circ - 360^\circ = -2x + 6x - 12^\circ
\][/tex]
Combine the constants and the [tex]\( x \)[/tex]-terms:
[tex]\[
180^\circ = 4x - 12^\circ
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Add [tex]\( 12^\circ \)[/tex] to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
180^\circ + 12^\circ = 4x
\][/tex]
[tex]\[
192^\circ = 4x
\][/tex]
Divide both sides by 4:
[tex]\[
x = \frac{192^\circ}{4} = 48^\circ
\][/tex]
7. Verify the result:
- The angle [tex]\( x \)[/tex] is [tex]\( 48^\circ \)[/tex].
- The complement of the angle is [tex]\( 90^\circ - 48^\circ = 42^\circ \)[/tex].
- The supplement of the angle is [tex]\( 180^\circ - 48^\circ = 132^\circ \)[/tex].
- Check the given condition:
- Six times the complement: [tex]\( 6 \times 42^\circ = 252^\circ \)[/tex].
- Twice the supplement minus 12 degrees: [tex]\( 2 \times 132^\circ - 12^\circ = 264^\circ - 12^\circ = 252^\circ \)[/tex].
Since both expressions equal [tex]\( 252^\circ \)[/tex], the solution holds true.
Therefore, the measure of the angle is [tex]\( 48^\circ \)[/tex].
