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