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Hi, this is Julie Harland and I'm your math gal.
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Please visit my website at yourmathgal.com where you could search for any of my videos organized by topic.
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This is Angles Part 4.
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In this video we learn more terms and definitions and learn some more symbols.
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And we work through two more problems.
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This is the first problem we do.
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And this is the second problem we do.
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So we're going to begin by coming up with a few more definitions and some symbols.
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First of all, parallel lines never meet.
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R and S below here parallel.
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And we use this symbol, the two vertical lines here, to mean as parallel to you.
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If I write R is parallel to S, that's what it looks like.
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Perpendicular lines are intersecting lines.
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They actually do intersect, but they intersect at right angles.
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So A and B are perpendicular.
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And we use this little symbol here, the sort of upside down T, to mean as perpendicular to.
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So this says A is perpendicular to B.
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And all these angles here are 90 degrees.
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So all four angles that you see here are 90 degree angles.
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In the previous video, we already discovered that vertical angles have the same measure.
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And that adjacent angles of two intersecting lines form supplementary angles.
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So in this picture here, these two intersecting lines, A and C are vertical angles.
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So they have the same measure, A equals C.
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B and D are vertical angles. So they have the same measure. So B equals D.
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And also, each group here next to each other, the set of adjacent angles, are supplementary angles.
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So A plus B, add up to 180 degrees.
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B plus C, add up to 180. C plus D, and A plus D.
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They all, each pair of those angles, adds up to 180 degrees.
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So if you have two intersecting lines, this is something we know about all the angles right off the bat.
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Okay, let's look at two parallel lines, S and R, and a transversal T.
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A transversal is a line that goes through two other lines.
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Okay, so we have two parallel lines and a transversal, I'll call that line T, going through it.
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You see that there are eight angles formed here.
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Now, when you look at all of these angles, or something interesting,
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there are A, for instance, is a large angle, where B is a smaller angle.
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See the difference? So you see four large angles, A, D, E, and H.
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And they actually all have the same measure.
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Now, corresponding angles are, if you look at this transversal here,
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you look on the same side of it, and they're both on top, A, and E, for instance,
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are called corresponding angles, or B, and F are called corresponding angles,
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or D, and H are called corresponding angles, et cetera.
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So hopefully, when you look at this picture, it makes sense to you that the way this line cuts through parallel lines
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that A and E should be the same measure.
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So I have that right here, A and E are corresponding angles, which have the same measure.
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We already talked about vertical angles before.
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There's a whole bunch of sets of vertical angles, A and D, or vertical angles, or B and C.
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I've written a couple, just one pair, E and H, right here, E and H.
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These are some vertical angles.
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Okay, there's a couple other terms here, alternate interior, and alternate exterior angles.
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Here's what an alternate interior angle is.
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If you look at the interior between the two parallel lines,
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you look at angles that are on the opposite side of each other,
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but kind of alternate from each other.
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So C and F are on opposite sides of this transversal, but they're alternate here, right?
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One's on the left side, and this one's on the right side, et cetera, below.
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So we say C and F are alternate interior angles, or I could say,
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what else, D and E are the other pair of alternate interior angles.
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How about X-terrier?
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We're looking at what's above one of the parallel lines, and below the other parallel line.
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So again, on opposite sides, G and B are alternate, but they're exterior,
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and A and H are alternate but exterior.
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So all of these angles, alternate interior angles,
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have the same measure, alternate exterior angles, have the same measure,
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corresponding angles have the same measure, and vertical angles have the same measure.
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But the main thing I want you to get here is that four of these angles are all the same.
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So watch, the big angles are all the same.
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A, D, E, and F are all the same.
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A, D, E, and F are all the larger of the two angles you basically see here.
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That is an obtuse angle, right?
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Those are all obtuse angles, and then we take all the smaller angles,
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and they're all equal to each other, B, C, F, and G.
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So B, C, oops, I think I said that wrong.
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This is what I did wrong.
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That was supposed to be E and H, there we go, sorry.
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So B, C, F, and G are the smaller angles.
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Okay, so try to take that in.
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When you have two parallel lines and a transversal going through them,
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that there are really eight angles total,
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but there's only two measures that you're going to see,
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the obtuse angle and the acute angle.
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And of course, if the transversal goes through a right angle,
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then of course there are all 90 degree angles.
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There is no obtuse and acute angle.
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All right, so we're going to try to take all that in and do a problem.
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If line 1 and line 2 are parallel, find the measure of angle A and B.
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All right, see if you could figure this out on your own first
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by putting the video on pause.
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Okay, let's go for it here.
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Trying to find A and B.
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Well, I know that A plus B add up to 180,
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but I have two variables there, so that doesn't quite help me yet.
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I know that this angle up here, the 72 degree angle,
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and B, those are alternate exterior angles.
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Those are the small angles, that's the acute angle.
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So, aha, right away I know what B is.
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It's the same as that one up here, 72 degrees.
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So I know this is 72 degrees, and that's half my answer.
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I remember I'm trying to find out what A is.
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Trying to find out what B is.
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I got 72 degrees right off the bat.
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Now I need to find A.
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That is the larger of the two angles, the obtuse one,
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and I know that these two angles together,
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A and B, add up to 180.
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So I know that A plus B is 180, right?
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But I know it B is, it's 72.
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So really, I've got A plus 72 is 180,
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and so we just subtract 72 from both sides.
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And I've got it.
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So now this is 108 degrees, and that's what I need to know.
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Do those add up to 180?
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They certainly do, and so there's a answer to that problem.
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Okay, one more problem.
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If line 1 and line 2 are parallel, find X.
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So again, we've got this transversal going through,
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and we're trying to find 2X and 7X.
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So see if you could figure this out on your own first.
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Okay, here's one way to think about it.
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You've got the small angle and the large angle, right?
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Remember, there's always an acute angle and an obtuse angle,
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and they're always supplementary angles.
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The smaller one plus the larger one adds up to 180 degrees.
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So 2X plus 7X has to be 180 degrees.
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Now maybe you don't think about it that way.
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Somebody else might have done.
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You said, well, if this is 7X, this angle down here is also 7X,
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because those are the corresponding ones.
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And then it's easy to see why these two really are supplementary angles,
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7X plus 2X, add up to 180.
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So I get the same equation.
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I think it'll battle a little differently.
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Now if I divide by 9, what would X be?
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20 degrees. So that's the first question, find X.
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Second part, find the measure of angle A.
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Okay, what's A?
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Oh, A's the same thing as 7X, right?
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A, because those are vertical angles,
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A's the same thing as 7X, and 7X is 7 times 20,
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or 140 degrees.
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And so those were the two things we were asked to do, and that would be our answer.
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Let's just look at this one more time.
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Remember of R is parallel to S.
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We came up with A, D, E, and H, where all the same measure,
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and B, C, F, and G were all the same measure.
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But the other thing is all these pairs of angles add up to 180.
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So like E and G add up to 180, G and H add up to 180,
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and H and F, etc.
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So basically if you take the smaller angle,
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that's what I just did in this last problem, the smaller angle,
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plus the larger angle,
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add up to 180.
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So I'm just going to put...
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I really mean the measure of the smaller angle,
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and the measure of the larger angle add up to 180.
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So I could write here D and F, for instance, add up to 180.
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You could take any of the obtuse angle, add it to any of the acute angle,
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and it would add up to 180.
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So if I took B, for instance,
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I can add it to one of the larger angles like H,
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and that should also be 180.
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Okay?
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So these are the larger angles, A, D, E, and H.
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These are the smaller angles,
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and any one of these added to any one of these is 180.
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They're supplementary angles.
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Please visit my website at yourmathgal.com
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where you can view all of my videos, which are organized by topic.