Introduction to angles (old) What an angle is. Angles in a circle. Complementary and supplementary angles.
Introduction to angles (old)
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- Hello. In this series of presentations, I'm gonna try
- to teach you everything you need to know about triangles and angles and parallel lines
- and this is probably the highest-yield information that you could ever learn, especially in terms of the standardized tests.
- And then when we've learned all the rules we'll play something I call the Angle Game,
- which is essentially what the SAT makes you do over and over again.
- So let's start with some basics.You know what an angles is.
- Well actually maybe you don't know what an angle is.
- If I have two lines...
- and they intersect at some point,
- the angle is a measure of exactly how wide the intersection is between those two lines.
- So this is the angle. An angle is how wide those two lines open up.
- And they're measured either in degrees or radiants. And for the sake of most geometry classes we'll use degrees.
- When we start doing Trigonometry we'll use radiants.
- And you're probably familiar with this. Zero degrees would be two lines on top of each other...
- this if I were to just eyeball it looks like 45 degrees.
- If I had the lines even wider apart, like that, that's 90 degrees.
- And 90 degree lines are also called perpendicular, because
- they are, I feel like saying because they are perpendicular,
- but because one is going completely vertical while the other is going horizontal.
- Wow, it's actually amazingly difficult to find the exact right wording.
- But I think you get the idea. By definition, perpendicular lines are 90 degrees apart from each other.
- And you've seen this all the time in things like squares or rectangles.
- A rectangle is made up of a bunch of perpendicular lines, or lines at 90 degree angles.
- The way you draw a 90 degree angle is you draw a little box like that.
- That's the same thing as doing this.
- And you could even get wider angles. If you go above 90 degrees... this could be, I don't know, 135 degrees
- If you ever want to really measure the angles you could use a protractor.
- Then if you had it so wide that the two lines are actually almost forming a line...
- that's 180 degrees. And then you could keep going.
- If this angle is 135 degrees...
- There are 360 degrees in a circle. So this magenta angle would be 360 - 135 degrees
- that's 225 degrees.
- So you know degrees in a circle are 360 degrees, this is important to know.
- It's also important to know that if you go halfway around a circle,
- that's 180 degrees.
- Like if you viewed the pivot point as like,
- let's say, right here.
- I mean it looks like just one line and it really is.
- But that's 180 degrees.
- And then if you go quarter way around the circle,
- that's 90 degrees.
- All right?
- Hopefully you're getting a bit of an intuition
- for what an angle is.
- So now I will teach you a bunch of very useful
- rules for angles.
- Clear this.
- So let me redraw.
- So if I had a line like this.
- I like using the colors, just so I think it keeps you from
- getting completely bored.
- And it might not be completely intuitive what I'm doing, but
- let's add an angle like that.
- And so, let's just say-- you know, I'm not measuring these
- exactly-- let's say that this is 30 degrees.
- We know that if we go all the way around the circle, we know
- that that's 360 degrees.
- And that's a very ugly looking around the circle
- angle that I drew.
- So then we also know that this angle right
- here is 330 degrees.
- Because this angle plus this magenta angle is going to
- equal the whole circle.
- So this is equal to 330 degrees.
- So remember that.
- The angles in a circle-- or there are 360
- degrees in a circle.
- I don't know if you remember.
- You probably don't.
- This was probably before you were born.
- But there used to be a game called 720, and it was a
- skateboarding game-- it was a video game.
- And the 720 was essentially you were trying to jump
- your skateboard and spin around twice.
- And that's 720 degrees.
- If you go around a circle twice that's 720 degrees.
- If you just jump and spin around once, you
- went 360 degrees.
- So you've probably heard this in just popular culture.
- But anyway.
- So 360 degrees in a circle.
- And you could imagine half a circle is 180 degrees.
- So the other important thing to realize is, like we said, if
- we go halfway around the circle it's 180 degrees.
- But if we have two angles that add up to that-- so let's say.
- I don't know if these lines are thick enough for you to see.
- Let me draw something thicker.
- It doesn't look ideal, but you get the idea.
- So if we have this angle, let's call it x.
- And then this angle is y.
- What do we know about the relationship between x and y?
- Well, we know that the entire angle is half of a circle.
- So that's 180 degrees.
- That's 180 degrees, this entire angle.
- So what are angles x and y going to add up to?
- I'm trying to stay color consistent.
- x plus y are going to equal-- I'm color blind,
- I think-- 180 degrees.
- Or you could write y is equal to 180 minus x.
- Or x is equal to 180 minus y.
- But if x plus y are equal to 180 degrees-- and you can see
- that it makes sense that they do-- if you add the two angles
- you go halfway around a circle.
- Then that tells us that x and y are-- and this is a fancy word,
- and it's just good to commit this to memory-- they are
- supplementary angles.
- That's when you add to 180 degrees.
- Now what if we had this situation.
- Oh my God, that was horrible.
- Let's say I had this situation.
- Let's see.
- I draw two perpendicular lines.
- So this is going a quarter way around the circle.
- All right.
- Let's say this entire angle here-- I'm drawing it really
- big-- that's 90 degrees.
- They're perpendicular.
- And now if I had two angles within that.
- So now if I have two angles here-- so let's say that this
- is x and this is y-- what do x and y add up to?
- Well, x plus y is 90.
- And we can say that x and y are complementary.
- And it's important to not get confused between the two.
- Just remember complementary means two angles add up to 90
- degrees, supplementary means that two angles add
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