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There are many kinds of angles, including adjacent angles, congruent
angles,
complementary angles, and supplementary angles. Understanding this terminology helps us
tackle more advanced geometrical concepts with greater confidence.
So what exactly *are* supplementary angles? What can they teach
us about geometry? Let's find out:

Angles are supplementary if they add up to a total of 180 degrees. And that's it! They do not need to be adjacent or part of the same figure, any two angles whose degree sum is 180 are considered a supplementary pair.

Because we know that all supplementary angles add up to 180 degrees, we also know that the two angles in a linear pair are also supplementary. Just consider the following image:

You might also think that in order for angles to be supplementary, they must be adjacent. But this just isn't the case. Consider the following image:

As you can see, angles 3 and 4 are supplementary even though they are not adjacent. Why? Because they add up to 180 degrees.

Now let's solve a few problems using our knowledge of supplementary angles:

If we know that two angles are supplementary and that one angle is twice the measure of the other, then can we find out the measure of both angles?

We know that the angle "a" is half the size of the other angle. The total degrees are 180. This means that we can create the following equation:

$a+2a=180$

We can also write this as $3a=180$ .

We know that $60\times 3=180$ , so we have the value for the smaller side:

$a=60\xb0$

We also know that the second angle is twice as big, so this means that the second angle is 120 degrees.

Let's try another example:

We know that angles P and Q are supplementary.

We also know that angle $P=2x+15$ and angle $Q=5x-38$ .

$P+Q=180$ .

In other words, $2x+15+15x-38=180$

If we combine the like terms, we get this:

$7x-23=180$

Next, we can simplify by adding 23 to both sides:

$7x=203$

If we divide 203 by 7, we get 29. This means that x = 29. With this value, we can solve the equations for angles P and Q:

$P=2\left(29\right)+15$ or $58+15$ .

Angle P is 73 degrees.

$Q=5\left(29\right)-38$ or $145-38$ .

Angle Q is 107 degrees.

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