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平行線與橫向線形成的角度 (英): 平行線成的角度
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- In this video we're going to think a little bit about
- parallel lines, and other lines that intersect the parallel
- lines, and we call those transversals.
- So first let's think about what a parallel or what
- parallel lines are.
- So one definition we could use, and I think that'll work well
- for the purposes of this video, are they're two lines that
- sit in the same plane.
- And when I talk about a plane, I'm talking about a, you can
- imagine a flat two-dimensional surface like this screen --
- this screen is a plane.
- So two lines that sit in a plane that never intersect.
- So this line -- I'll try my best to draw it -- and imagine
- the line just keeps going in that direction and that
- direction -- let me do another one in a different color --
- and this line right here are parallel.
- They will never intersect.
- If you assume that I drew it straight enough and that
- they're going in the exact same direction, they
- will never intersect.
- And so if you think about what types of lines are not
- parallel, well, this green line and this pink line
- are not parallel.
- They clearly intersect at some point.
- So these two guys are parallel right over here, and sometimes
- it's specified, sometimes people will draw an arrow going
- in the same direction to show that those two lines
- are parallel.
- If there are multiple parallel lines, they might do two arrows
- and two arrows or whatever.
- But you just have to say OK, these lines will
- never intersect.
- What we want to think about is what happens when
- these parallel lines are intersected by a third line.
- Let me draw the third line here.
- So third line like this.
- And we call that, right there, the third line that intersects
- the parallel lines we call a transversal line.
- Because it tranverses the two parallel lines.
- Now whenever you have a transversal crossing parallel
- lines, you have an interesting relationship between
- the angles form.
- Now this shows up on a lot of standardized tests.
- It's kind of a core type of a geometry problem.
- So it's a good thing to really get clear in our heads.
- So the first thing to realize is if these lines are parallel,
- we're going to assume these lines are parallel, then we
- have corresponding angles are going to be the same.
- What I mean by corresponding angles are I guess you could
- think there are four angles that get formed when this
- purple line or this magenta line intersects
- this yellow line.
- You have this angle up here that I've specified in green,
- you have -- let me do another one in orange -- you have this
- angle right here in orange, you have this angle right here in
- this other shade of green, and then you have this angle
- right here -- right there that I've made in that
- bluish-purplish color.
- So those are the four angles.
- So when we talk about corresponding angles, we're
- talking about, for example, this top right angle in green
- up here, that corresponds to this top right angle in, what
- I can draw it in that same green, right over here.
- These two angles are corresponding.
- These two are corresponding angles and they're
- going to be equal.
- These are equal angles.
- If this is -- I'll make up a number -- if this is 70
- degrees, then this angle right here is also
- going to be 70 degrees.
- And if you just think about it, or if you even play with
- toothpicks or something, and you keep changing the direction
- of this transversal line, you'll see that it actually
- looks like they should always be equal.
- If I were to take -- let me draw two other parallel
- lines, let me show maybe a more extreme example.
- So if I have two other parallel lines like that, and then let
- me make a transversal that forms a smaller -- it's even a
- smaller angle here -- you see that this angle right here
- looks the same as that angle.
- Those are corresponding angles and they will be equivalent.
- From this perspective it's kind of the top right angle and each
- intersection is the same.
- Now the same is true of the other corresponding angles.
- This angle right here in this example, it's the top left
- angle will be the same as the top left angle right over here.
- This bottom left angle will be the same down here.
- If this right here is 70 degrees, then this down here
- will also be 70 degrees.
- And then finally, of course, this angle and this angle
- will also be the same.
- So corresponding angles -- let me write these -- these are
- corresponding angles are congruent.
- Corresponding angles are equal.
- And that and that are corresponding, that and
- that, that and that, and that and that.
- Now, the next set of equal angles to realize are sometimes
- they're called vertical angles, sometimes they're called
- opposite angles.
- But if you take this angle right here, the angle that is
- vertical to it or is opposite as you go right across the
- point of intersection is this angle right here, and that is
- going to be the same thing.
- So we could say opposite -- I like opposite because it's not
- always in the vertical direction, sometimes it's in
- the horizontal direction, but sometimes they're referred
- to as vertical angles.
- Opposite or vertical angles are also equal.
- So if that's 70 degrees, then this is also 70 degrees.
- And if this is 70 degrees, then this right here
- is also 70 degrees.
- So it's interesting, if that's 70 degrees and that's 70
- degrees, and if this is 70 degrees and that is also 70
- degrees, so no matter what this is, this will also be the same
- thing because this is the same as that, that
- is the same as that.
- Now, the last one that you need to I guess kind of realize are
- the relationship between this orange angle and this
- green angle right there.
- You can see that when you add up the angles, you go halfway
- around a circle, right?
- If you start here you do the green angle, then
- you do the orange angle.
- You go halfway around the circle, and that'll give you,
- it'll get you to 180 degrees.
- So this green and orange angle have to add up to 180 degrees
- or they are supplementary.
- And we've done other videos on supplementary, but you just
- have to realize they form the same line or a half circle.
- So if this right here is 70 degrees, then this orange angle
- right here is 110 degrees, because they add up to 180.
- Now, if this character right here is 110 degrees, what
- do we know about this character right here?
- Well, this character is opposite or vertical
- to the 110 degrees so it's also 110 degrees.
- We also know since this angle corresponds with this angle,
- this angle will also be 110 degrees.
- Or we could have said that look, because this is 70 and
- this guy is supplementary, these guys have to add up to
- 180 so you could have gotten it that way.
- And you could also figure out that since this is 110, this
- is a corresponding angle, it is also going to be 110.
- Or you could have said this is opposite to
- that so they're equal.
- Or you could have said that this is supplementary with
- that angle, so 70 plus 110 have to be 180.
- Or you could have said 70 plus this angle are 180.
- So there's a bunch of ways to come to figure out
- which angle is which.
- In the next video I'm just going to do a bunch of examples
- just to show that if you know one of these angles, you
- can really figure out all of the angles.
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