Question

If three times the supp. of an angle is subtracted from seven times the comp. of the angle, the answer is the same a that obtained by trisecting a right angle. Find the supplement.

Answer 1

Alright, let's start by giving some notation to "an angle." We'll use the Greek letter $\theta$ to denote it.

Next, we'll need to review a few definitions:

- Two supplementary angles add up to a straight angle, or 180°
- Two complementary angles add up to a right angle, or 90°
- Trisecting an angle is splitting a larger angle into three equal smaller angles.

If we call $\theta$'s supplement $\beta$ and $\theta$'s complement $\alpha$, we know that:

$\theta+\beta=180\\ \theta+\alpha=90$

or, if we want to put everything in terms of $\theta$:

$\beta=180-\theta\\\alpha=90-\theta$

We're given from the problem that $3\beta$ is being subtracted from $7\alpha$, which in terms of $\theta$ gives us:

$7(90-\theta)-3(180-\theta)$

Next, we're told that this expression is equal to the angle obtained by trisecting a right angle. A right angle is equal to 90°, so trisecting it, we get the angle 90°/3 = 30°.

Putting everything together, we have:

$7(90-\theta)-3(180-\theta)=30$

From there, solve for $\theta$, but remember that the question asks for the supplement of $\theta$, not $\theta$ itself. Fortunately, we have an equation from earlier for the supplement, $\beta$:

$\beta=180-\theta$

Simply put your result from solving for $\theta$ into that equation and solve, and you'll have your answer.