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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.