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In this video we're going to[br]think a little bit about

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parallel lines, and other lines[br]that intersect the parallel

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lines, and we call[br]those transversals.

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So first let's think about[br]what a parallel or what

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parallel lines are.

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So one definition we could use,[br]and I think that'll work well

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for the purposes of this video,[br]are they're two lines that

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sit in the same plane.

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And when I talk about a plane,[br]I'm talking about a, you can

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imagine a flat two-dimensional[br]surface like this screen --

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this screen is a plane.

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So two lines that sit in a[br]plane that never intersect.

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So this line -- I'll try my[br]best to draw it -- and imagine

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the line just keeps going in[br]that direction and that

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direction -- let me do another[br]one in a different color --

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and this line right[br]here are parallel.

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They will never intersect.

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If you assume that I drew it[br]straight enough and that

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they're going in the exact[br]same direction, they

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will never intersect.

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And so if you think about what[br]types of lines are not

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parallel, well, this green line[br]and this pink line

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are not parallel.

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They clearly intersect[br]at some point.

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So these two guys are parallel[br]right over here, and sometimes

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it's specified, sometimes[br]people will draw an arrow going

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in the same direction to show[br]that those two lines

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are parallel.

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If there are multiple parallel[br]lines, they might do two arrows

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and two arrows or whatever.

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But you just have to say[br]OK, these lines will

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never intersect.

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What we want to think about[br]is what happens when

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these parallel lines are[br]intersected by a third line.

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Let me draw the[br]third line here.

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So third line like this.

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And we call that, right there,[br]the third line that intersects

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the parallel lines we[br]call a transversal line.

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Because it tranverses[br]the two parallel lines.

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Now whenever you have a[br]transversal crossing parallel

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lines, you have an interesting[br]relationship between

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the angles form.

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Now this shows up on a lot[br]of standardized tests.

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It's kind of a core type[br]of a geometry problem.

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So it's a good thing to really[br]get clear in our heads.

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So the first thing to realize[br]is if these lines are parallel,

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we're going to assume these[br]lines are parallel, then we

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have corresponding angles[br]are going to be the same.

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What I mean by corresponding[br]angles are I guess you could

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think there are four angles[br]that get formed when this

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purple line or this[br]magenta line intersects

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this yellow line.

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You have this angle up here[br]that I've specified in green,

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you have -- let me do another[br]one in orange -- you have this

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angle right here in orange, you[br]have this angle right here in

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this other shade of green, and[br]then you have this angle

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right here -- right there[br]that I've made in that

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bluish-purplish color.

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So those are the four angles.

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So when we talk about[br]corresponding angles, we're

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talking about, for example,[br]this top right angle in green

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up here, that corresponds to[br]this top right angle in, what

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I can draw it in that same[br]green, right over here.

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These two angles[br]are corresponding.

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These two are corresponding[br]angles and they're

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going to be equal.

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These are equal angles.

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If this is -- I'll make up[br]a number -- if this is 70

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degrees, then this angle[br]right here is also

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going to be 70 degrees.

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And if you just think about it,[br]or if you even play with

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toothpicks or something, and[br]you keep changing the direction

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of this transversal line,[br]you'll see that it actually

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looks like they should[br]always be equal.

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If I were to take -- let me[br]draw two other parallel

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lines, let me show maybe[br]a more extreme example.

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So if I have two other parallel[br]lines like that, and then let

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me make a transversal that[br]forms a smaller -- it's even a

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smaller angle here -- you see[br]that this angle right here

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looks the same as that angle.

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Those are corresponding angles[br]and they will be equivalent.

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From this perspective it's kind[br]of the top right angle and each

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intersection is the same.

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Now the same is true of the[br]other corresponding angles.

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This angle right here in this[br]example, it's the top left

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angle will be the same as the[br]top left angle right over here.

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This bottom left angle will[br]be the same down here.

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If this right here is 70[br]degrees, then this down here

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will also be 70 degrees.

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And then finally, of course,[br]this angle and this angle

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will also be the same.

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So corresponding angles -- let[br]me write these -- these are

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corresponding angles[br]are congruent.

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Corresponding angles are equal.

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And that and that are[br]corresponding, that and

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that, that and that,[br]and that and that.

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Now, the next set of equal[br]angles to realize are sometimes

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they're called vertical angles,[br]sometimes they're called

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opposite angles.

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But if you take this angle[br]right here, the angle that is

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vertical to it or is opposite[br]as you go right across the

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point of intersection is this[br]angle right here, and that is

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going to be the same thing.

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So we could say opposite -- I[br]like opposite because it's not

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always in the vertical[br]direction, sometimes it's in

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the horizontal direction, but[br]sometimes they're referred

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to as vertical angles.

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Opposite or vertical[br]angles are also equal.

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So if that's 70 degrees, then[br]this is also 70 degrees.

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And if this is 70 degrees,[br]then this right here

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is also 70 degrees.

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So it's interesting, if that's[br]70 degrees and that's 70

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degrees, and if this is 70[br]degrees and that is also 70

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degrees, so no matter what this[br]is, this will also be the same

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thing because this is[br]the same as that, that

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is the same as that.

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Now, the last one that you need[br]to I guess kind of realize are

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the relationship between[br]this orange angle and this

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green angle right there.

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You can see that when you add[br]up the angles, you go halfway

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around a circle, right?

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If you start here you do[br]the green angle, then

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you do the orange angle.

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You go halfway around the[br]circle, and that'll give you,

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it'll get you to 180 degrees.

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So this green and orange angle[br]have to add up to 180 degrees

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or they are supplementary.

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And we've done other videos on[br]supplementary, but you just

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have to realize they form the[br]same line or a half circle.

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So if this right here is 70[br]degrees, then this orange angle

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right here is 110 degrees,[br]because they add up to 180.

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Now, if this character right[br]here is 110 degrees, what

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do we know about this[br]character right here?

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Well, this character is[br]opposite or vertical

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to the 110 degrees so[br]it's also 110 degrees.

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We also know since this angle[br]corresponds with this angle,

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this angle will also[br]be 110 degrees.

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Or we could have said that[br]look, because this is 70 and

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this guy is supplementary,[br]these guys have to add up to

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180 so you could have[br]gotten it that way.

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And you could also figure out[br]that since this is 110, this

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is a corresponding angle,[br]it is also going to be 110.

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Or you could have said[br]this is opposite to

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that so they're equal.

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Or you could have said that[br]this is supplementary with

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that angle, so 70 plus[br]110 have to be 180.

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Or you could have said 70[br]plus this angle are 180.

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So there's a bunch of ways[br]to come to figure out

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which angle is which.

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In the next video I'm just[br]going to do a bunch of examples

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just to show that if you know[br]one of these angles, you

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can really figure out[br]all of the angles.