Complementary and Supplementary Angles

Basics of complementary, supplementary, adjacent and straight angles. Also touching on what it means to be perpendicular

Let's say I have an angle ABC, and it looks somethings like this, so its vertex is going to be at 'B',

Maybe 'A' sits right over here, and 'C' sits right over there.

And then also let's say we have another angle called DAB, actually let me call it DBA,

I want to have the vertex once again at 'B'.

So let's say it looks like this, so this right over here is our point 'D'.

And let's say we know the measure of angle DBA, let's say we know that that's equal to 40 degrees.

So this angle right over here, its measure is equal to 40 degrees,

And let's say we know that the measure of angle ABC is equal to 50 degrees.

Right, so there's a bunch of interesting things happening over here,

the first interesting thing that you might realize is that both of these angles

share a side, if you view these as rays, they could be lines,

line segments or rays, but if you view them as rays,

then they both share the ray BA, and when you have two angles

like this that share the same side, these are called adjacent angles

because the word adjacent literally means 'next to'.

Adjacent, these are adjacent angles.

Now there's something else you might notice that's interesting here,

we know that the measure of angle DBA is 40 degreees

and the measure of angle ABC is 50 degrees

and you might be able to guess what the measure of angle DBC is,

the measure of angle DBC, if we drew a protractor over here

I'm not going to draw it, it will make my drawing all messy,

but if we, well I'll draw it really fast,

So, if we had a protractor over here, clearly this is opening up to 50 degrees,

and this is going another 40 degrees, so if you wanted to say

what the measure of angle DBC is,

it would be, it would essentially be the the sum of 40 degrees and 50 degrees.

And let me delete all this stuff right here, to keep things clean,

So the measure of angle DBC would be equal to 90 degrees

and we already know that 90 degrees is a special angle,

this is a right angle, this is a right angle.

There's also a word for two angles whose sum add to 90 degrees,

and that is complementary.

So we can also say that angle DBA and angles ABC are complementary.

And that is because their measures add up to 90 degrees,

So the measure of angle DBA plus the measure of angle ABC,

is equal to 90 degrees, they form a right angle when you add them up.

And just as another point of terminology, that's kind of related to right angles,

when you form, when a right angle is formed, the two rays that form the right angle,

or the two lines that form that right angle, or the two line segments,

are called perpendicular.

So because we know the measure of angle DBC is 90 degrees,

or that angle DBC is a right angle, this tells us

that DB, if I call them, maybe the line segment DB is

perpendicular, is perpendicular to line segment BC,

or we could even say that ray BD, is instead of using the word perpendicular

there is sometimes this symbol right here, which just shows two perpendicular lines,

DB is perpendicular to BC

So all of these are true statements here,

and these come out of the fact that the angle formed between DB and BC

that is a 90 degree angle.

Now we have other words when our two angles add up to other things,

so let's say for example I have one angle over here,

that is, I'll just make up, let's just call this angle,

let me just put some letters here to specify, 'X', 'Y' and 'Z'.

Let's say that the measure of angle XYZ is equal to 60 degrees,

and let's say you have another angle, that looks like this,

and I'll call this, maybe 'M', 'N', 'O',

and let's say that the measure of angle MNO is 120 degrees.

So if you were to add the two measures of these, so let me write this down,

the measure of angle MNO plus the measure of angle XYZ,

is equal to, this is going to be equal to 120 degrees plus 60 degrees.

Which is equal to 180 degrees, so if you add these two things up,

you're essentially able to go halfway around the circle.

Or throughout the entire half circle, or a semi-circle for a protractor.

And when you have two angles that add up to 180 degrees, we call them supplementary angles

I know it's a little hard to remember sometimes, 90 degrees is complementary,

there are two angles complementing each other,

and then if you add up to 180 degrees, you have supplementary angles,

and if you have two supplementary angles that are adjacent,

so they share a common side, so let me draw that over here,

So let's say you have one angle that looks like this,

And that you have another angle, so so let me put some letters here again,

and I'll start re-using letters,

so this is 'A', 'B', 'C', and you have another angle that looks like this,

that looks like this, I already used 'C', that looks like this

notice and let's say once again that this is 50 degrees,

and this right over here is 130 degrees,

clearly angle DBA plus angle ABC, if you add them together,

you get 180 degrees.

So they are supplementary, let me write that down,

Angle DBA and angle ABC are supplementary,

they add up to 180 degrees, but they are also adjacent angles,

they are also adjacent, and because they are supplementary and they're adjacent,

if you look at the broader angle, the angle formed from the sides they don't have in common,

if you look at angle DBC, this is going to be essentially a straight line,

which we can call a straight angle.

So I've introduced you to a bunch of words here and now I think

we have all of the tools we need to start doing some interesting proofs,

and just to review here we talked about adjacent angles, and I guess any angles

that add up to 90 degrees are considered to be complementary,

this is adding up to 90 degrees.

If they happen to be adjacent then the two outside sides will form a right angle,

when you have a right angle the two sides of a right angle are considered to be

perpendicular.

And then if you have two angles that add up 180 degrees,

they are considered supplementary, and then if they happen to be adjacent,

they will form a straight angle.

Or another way of saying itis that if you have a straight angle,

and you have one of the angles, the other angle

is going to be supplementary to it, they're going to add up to 180 degrees.

So I'll leave you there.