In order for two angles to be complementary their sum must equal \(\displaystyle 90^{\circ}\), therefore we subtract the given angle from \(\displaystyle 90^{\circ}\) as follows:

\(\displaystyle 90^{\circ} - 100^{\circ} = -10^{\circ}\)

However, it was stated that we are looking for the positive complementary angle, therefore, there is no positive complementary angle.

If two angles are supplementary their sum must be \(\displaystyle \pi\), therefore we subtract the given angle from \(\displaystyle \pi\) to solve for the missing angle as follows:

\(\displaystyle \pi -0 = x\)

\(\displaystyle x=\pi\)

For angles to be complement to each other, the two angle sets must sum to 90 degrees or \(\displaystyle \frac{\pi}{2}\) radians.

Since all the sets of angles given add up to either of the two possibilites given, all of the angle sets are complement to each other.

\(\displaystyle 89+1=90\)

\(\displaystyle 46+44=90\)

\(\displaystyle \frac{3\pi}{8}+\frac{\pi}{8}=\frac{4\pi}{8}=\frac{\pi}{2}\)

The correct answer is:

All sets of angles given are complement to each other.

The supplementary angles must add up to 180 degrees.  

In this case, to find the other angle supplementary to 15 degrees, simply subtract this angle from 180 degrees.

\(\displaystyle 180-15=165 \textup{ degrees}\)

If \(\displaystyle \small sin(\theta)\) is \(\displaystyle \small \frac{1}{2}\), that means that \(\displaystyle \theta\) is either \(\displaystyle \small \frac{\pi}{6}\) or \(\displaystyle \small \frac{5\pi}{6}\). A complementary angle will add to \(\displaystyle \frac{\pi}{2}\) with \(\displaystyle \small \theta\), so we can find what the complement is by subtracting our potential \(\displaystyle \small \theta\)s from \(\displaystyle \small \frac{\pi}{2}\), or equivalently, \(\displaystyle \small \frac{{\color{DarkOrange} 3*}\pi}{{\color{DarkOrange} 3*}2} = \frac{3\pi}{6}\):

\(\displaystyle \small \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6 }= \frac{\pi}{3}\)

\(\displaystyle \frac{3\pi}{6} - \frac{5\pi}{6} = \frac{-2\pi}{6} = \frac{-\pi}{3}\)Since all of the answer choices are positive, we want to see what the positive equivalent of this angle is. We can do this by adding \(\displaystyle 2\pi\), or equivalently \(\displaystyle \frac{6\pi}{3}\):

\(\displaystyle \small \frac{-\pi}{3}+\frac{6\pi}{3} = \frac{5\pi}{3}\)

This does not appear as an answer choice, but \(\displaystyle \small \frac{\pi}{3}\) does.

In order to answer this question, we need to know that supplementary angles add together to be \(\displaystyle 2\pi\). We could use this to create an algebraic equation:

\(\displaystyle x + x = 2\pi\) combine like terms

\(\displaystyle \small 2 x = 2 \pi\) divide both sides by 2

\(\displaystyle \small x = \pi\)

We could also do this math mentally. If the angle plus itself equals \(\displaystyle 2\pi\), then we can divide \(\displaystyle \small 2\pi\) by 2 to get \(\displaystyle \pi\).

Two angles are complementary if they add to \(\displaystyle \small \frac{\pi}{2}\). There are several ways of setting up this problem, but here is one. We can consider the angle we're looking for as "x."

If the angle we are looking for, x, is half its complement, that means that its complement is twice the angle, or 2x. We can then set up this equation to solve for x:

\(\displaystyle x + 2x = \frac{\pi}{2}\)  combine like terms

\(\displaystyle 3x = \frac{\pi}{2}\) divide both sides by 3, or equivalently multiply by \(\displaystyle \frac{1}{3}\)

\(\displaystyle x = \frac{\pi}{2} * \frac{ 1}{3 } = \frac{ \pi }{6 }\)

Complementary angles add to \(\displaystyle \frac{ \pi }{2}\).

Among the choices, this includes \(\displaystyle -\frac{ 7 \pi }{4}\) since \(\displaystyle \frac{ \pi }{4} - \frac{ 7 \pi }{4} = \frac{ - 6 \pi }{4 } = \frac {-3 \pi }{ 2 }\)

As a positive angle, this is \(\displaystyle \frac{ \pi }{2}\).

It includes \(\displaystyle \frac{ \pi }{4}\) since \(\displaystyle \frac{ \pi }{4} + \frac{ \pi }{4} = \frac{ \pi }{2}\)

Supplementary angles add to \(\displaystyle \pi\).

Among the choices, this includes \(\displaystyle \frac{ 3 \pi }{4}\) since \(\displaystyle \frac{ 3 \pi }{4} + \frac{ \pi }{4} = \frac{ 4 \pi }{4} = \pi\)

It also includes \(\displaystyle \frac{ 11 \pi }{4}\) since \(\displaystyle \frac{ 11 \pi }{4} + \frac{ \pi }{4} = \frac{ 12 \pi }{4} = 3 \pi\) which is coterminal with \(\displaystyle \pi\).

The one that doesn't work is \(\displaystyle \frac{ 7 \pi }{4}\) since \(\displaystyle \frac{ 7 \pi }{4} + \frac{ \pi }{4} = \frac{ 8 \pi }{4} = 2 \pi\).

In order for angles to be supplementary, they need to add to \(\displaystyle \pi\), or an angle coterminal with \(\displaystyle \pi\).

All of these work except for \(\displaystyle \frac{ 3 \pi }{10 }\):

1) \(\displaystyle \frac{ \pi }{5} + \frac{ 24 \pi }{5} = \frac{ 25 \pi }{5} = 5 \pi\)

We can see that \(\displaystyle 5 \pi\) is coterminal with \(\displaystyle \pi\) by subtracting \(\displaystyle 2\pi\): \(\displaystyle 5 \pi - 2 \pi = 3 \pi - 2 \pi = \pi\)

2) \(\displaystyle \frac{ 4 \pi }{5} + \frac{ \pi }{5} = \frac{ 5 \pi }{5} = \pi\)

3) \(\displaystyle \frac{ -6 \pi }{5} + \frac{ \pi }{5} = \frac{ - 5 \pi }{5} = - \pi\)

Add \(\displaystyle 2 \pi\) to determine the positive coterminal angle: \(\displaystyle - \pi + 2 \pi = \pi\)

4) \(\displaystyle - \frac{ 16 \pi }{5} + \frac{ \pi }{5} = \frac{ -15 \pi }{5} = - 3 \pi\)

Add \(\displaystyle 2 \pi\) to determine the positive coterminal angle: \(\displaystyle - 3 \pi + 2 \pi = - \pi + 2 \pi = \pi\)

5) \(\displaystyle \frac{ 3 \pi }{10 } + \frac{ \pi }{5} = \frac{ 3 \pi }{10 } + \frac{ 2 \pi }{10 } = \frac{ 5 \pi }{10 } = \frac{ \pi }{2}\) not \(\displaystyle \pi\)

In order to be supplementary, two angles must add to \(\displaystyle \pi\) or a coterminal angle. All of these work except for \(\displaystyle \frac{ 11 \pi }{7 }\):

1) \(\displaystyle -\frac{ 24 \pi }{7 } + \frac{ 3 \pi }{7 } = \frac{ -21 \pi }{7 } = -3 \pi\)

We can determine the positive coterminal angle by adding \(\displaystyle 2 \pi\):

\(\displaystyle - 3\pi + 2 \pi = - \pi + 2 \pi = \pi\)

2) \(\displaystyle - \frac{ 10 \pi }{7 } + \frac{ 3 \pi }{7 } = \frac{ - 7 \pi }{ 7 } = - \pi + 2\pi = \pi\)

3) \(\displaystyle \frac{ 18 \pi }{7 } + \frac{ 3 \pi }{7 } = \frac{ 21 \pi }{7 } = 3 \pi - 2 \pi = \pi\)

4) \(\displaystyle \frac{ 4 \pi }{ 7 } + \frac{ 3 \pi }{7 } = \frac{ 7 \pi }{7 } = \pi\)

5) \(\displaystyle \frac{ 11 \pi }{ 7 } + \frac{ 3 \pi }{7 } = \frac{ 14 \pi }{7 } = 2 \pi \neq \pi\)