Angles (part 2) More on complementary and supplementary angles. Introduction to opposite angles.
Angles (part 2)
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- So let's review everything that we know so far, because it's
- good to keep reviewing.
- Because these are things you should never forget
- the rest of your life.
- So if I have a line and if I draw an angle that goes--
- let's say this is the pivot point, right?
- If I go all the way around the line, or in a circle,
- that's 360 degrees.
- 11 00:00:23,97 --> 00:00:27,43 We learned that there are 360 degrees in a circle.
- We also learned that if I have lines like this.
- If I have two angles-- let me draw it like that.
- 16 00:00:43,71 --> 00:00:49,49 And this is angle x.
- This is angle y.
- 19 00:00:54,122 --> 00:00:57,825 x and y are supplementary.
- 21 00:01:04,24 --> 00:01:07,63 And that just means that they add up to 180 degrees.
- x plus y is equal to 180 degrees.
- And why does that make sense?
- Because look, if we add up x plus y we have gone
- halfway around the circle.
- So that's 180 degrees, right?
- So hopefully we have learned that.
- And then let me switch colors for the sake of variety.
- Let me use my line tool.
- If I have-- let's see, I'm going to draw
- perpendicular lines.
- If I have that line, and then I have that line.
- And they are perpendicular.
- And then I have another line.
- Let's say it goes like that.
- 39 00:01:59,83 --> 00:02:03,73 And then I say that this is angle x.
- This is angle x.
- And this is angle y.
- 44 00:02:12,29 --> 00:02:16,04 Well, I said this line and this line are perpendicular, right?
- So that means that they intersect at a 90 degree angle.
- So we know that this whole thing is 90 degrees.
- 48 00:02:24,07 --> 00:02:26,01 And so what do we know about x plus y?
- Well, x plus y is going to equal 90 degrees.
- 51 00:02:34,13 --> 00:02:41,33 Or we could say that x and y are complementary.
- And I always get confused between supplementary
- and complementary.
- You just got to memorize it.
- I don't know if there's any-- let's see, is
- there any easy way?
- 180, supplementary.
- You could say that 180-- 100 starts with an O, which
- supplementary does not start with.
- So there.
- There's your mnemonic.
- And 90 starts with an N, and complementary does
- not start with an N.
- That's your other mnemonic.
- 68 00:03:15,42 --> 00:03:16,59 I don't know if I'm spelling it right.
- Who cares?
- Let's move on.
- So let's learn some more stuff about angles.
- And what I'm going to do is I'm going to give you an arsenal,
- and then once you have that arsenal you can just tackle
- these beastly problems that I'm going to throw at you.
- So just take these for granted right now, and then in a few
- videos, probably, we're going to tackle some
- beastly problems.
- 79 00:03:38,22 --> 00:03:40,48 And you know, I'm using variables here.
- And if you're not familiar with variables you
- can put numbers here.
- If x was 30 degrees, then y is going to be 60 degrees.
- Or in this case, if x is, I don't know, 45 degrees, then y
- is going to be 135 degrees.
- That other way.
- Let me draw another property of angles of intersecting lines.
- So if I have two angles, two lines that intersect like this.
- 90 00:04:08,56 --> 00:04:10,76 So a couple of interesting things.
- So first, I'm going to teach you about opposite angles.
- 93 00:04:17,49 --> 00:04:19,51 Let me switch colors.
- Let me switch to yellow.
- So if this is x degrees, then it turns out that the angle
- opposite to it is also equal to x degrees.
- 98 00:04:40,43 --> 00:04:42,18 And you don't believe me?
- Well let me prove it to you.
- Let's say we call this, I don't know, let's
- call this y degrees.
- And I'm going to prove to you that the x and
- the y are the same.
- Well what do we know already?
- Let's call this other angle-- and I'm doing this to
- confuse you-- angle z.
- Well what do we know about angle x and angle z?
- It may not be obvious to you because I've drawn it slightly
- different, but I'll give you a small hint with
- an appropriately interesting color.
- So what angle is this whole thing right here?
- Well I'm just going along a line, right?
- That's halfway around a circle.
- So what does x plus z equal?
- Well, x plus z is going to equal that larger angle.
- x plus purple z is going to equal-- I think I'll switch to
- the blue; maybe it's taking too much time for me to switch--
- is equal to 180 degrees.
- Or x and z are supplementary.
- 124 00:06:09,24 --> 00:06:10,63 I've run out of space.
- So what do we know about z?
- Well z is equal to 180 minus x.
- Because x plus z is 180.
- Now, what's the relationship between z and y?
- Well, z and y are also supplementary.
- Because look, if I drew this angle here.
- Look at this big angle.
- 135 00:06:42,07 --> 00:06:42,69 What angle is that?
- Well once again I'm still going halfway around the circle.
- But now I'm using this line right here.
- So that's 180 degrees.
- So we know that angle z plus angle y is also
- equal to 180 degrees.
- 143 00:07:05,54 --> 00:07:06,82 Right?
- Or, I don't want to keep writing it, but z and y
- are also supplementary.
- But we just figured out that z is 180 minus x.
- So let's just substitute that back in here.
- So we get 180 minus x plus y is equal to 180 degrees.
- Why don't we subtract 180 degrees from both sides
- of this equation.
- That cancels out, and we get minus x plus y is equal to 0.
- And then add x to both sides of this equation, and
- we get y is equal to x.
- So x is equal to y.
- And if you've played around with this, if you just drew a
- bunch of straight lines and they intersected at different
- angles, I think when you eyeball it it would make sense.
- And then similarly, if that's z then the other opposite angle
- here is also z degrees.
- So what do we know now?
- The total angles in a circle, 360 degrees.
- When two angles kind of combine, go halfway around the
- circle-- or they combine, kind of form a line.
- There's different ways you can think about it.
- We know they're supplementary.
- They add up to 180 degrees.
- x plus y is 180 degrees.
- If they add up to 90 it's complementary.
- x plus y is 90.
- And then opposite angles are equal to each other.
- This angle is equal to this angle.
- And then this angle is going to be equal to this angle for the
- same reason-- because it's opposite.
- In the next video I'm going to show you about parallel
- lines and transversals.
- More fancy words for what I think are fairly
- straightforward concepts.
- I'll see you in the next video.
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