Complementary & supplementary angles

Let's say I have an angle ABC, and it looks something 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 that we have another angle called 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. That is our point D. And let's say that we know that the measure of angle DBA is equal to 40 degrees. So this angle right over here, its measure is equal to 40 degrees. And let's say that we know that the measure of angle ABC is equal to 50 degrees. So there's a bunch of interesting things happening 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, they both share this 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. These are adjacent. They are adjacent angles. Now there's something else that you might notice that's interesting here. We know that the measure of angle DBA is 40 degrees and the measure of angle ABC is 50 degrees. And you might be able to guess what the measure of angle DBC is. If we drew a protractor over here-- I'm not going to draw it. It will make my drawing all messy. Well, maybe I'll draw it really fast. So if you had a protractor right 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 essentially be the sum of 40 degrees and 50 degrees. And let me delete all of 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. There's also a word for two angles whose sum add up to 90 degrees, and that is complementary. So we can also say that angles 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 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 that form that right angle are called perpendicular. So because we know that measure of angle DBC is 90 degrees or that angle DBC is a right angle, this tells us, we know that the line segment DB is perpendicular to line segment BC. Or we could even say ray BD is-- instead of using the word perpendicular, there's sometimes this symbol right here, which really just shows two perpendicular lines-- 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. Let me put some letters here so we can specify it. So let's say this is X, Y, and Z. And let's say that the measure of angle XYZ is equal to 60 degrees. And let's say that you have another angle that looks like this. And I'll call this, let's say, maybe MNO. 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, 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 essentially are able to go all halfway around the circle. Or you could go throughout the entire half circle or semicircle for a protractor. And when you have two angles that add up to 180 degrees, we call them supplementary. I know it's a little hard to remember sometimes. 90 degrees is complementary. They're just complementing each other. And then if you add up to 180 degrees, you have supplementary. You have supplementary angles. And if you have two supplementary angles that are adjacent so that 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 let me put some letters here again. And I'll start reusing letters. So let's say that this is ABC. And you have another angle that looks like this. I already used C. Once again, let's say that this is 50 degrees. And let's say that this right over here is 130 degrees. Clearly, angle DBA plus angle ABC, if you add them together, you get 130 degrees plus 50 degrees, which is 180 degrees. So they are supplementary. So let me write that down. Angle DBA and angle ABC are supplementary. They add up to 180 degrees. But they are also adjacent angles. And because they're supplementary and they're adjacent, if you look at the broader angle, the angle used from the sides that 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 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 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 to 180 degrees, they're considered supplementary. And then if they happen to be adjacent, they will form a straight angle. Or another way, if you said, 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.