WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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The next lesson is on angles and parallel lines.
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OK, last lesson, we learned about the different special angle relationships, when we have a transversal.
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The transversal with the other lines forms angles, and those pairs of angles have special relationships.
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And one of them was the corresponding angles.
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Now, the two lines that the transversal cuts through--remember: I said that the lines can be parallel, but they don't have to be.
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So, even if the lines are not parallel, you are still going to have corresponding angles.
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But then, now the postulate is saying that, if the lines are parallel, then the corresponding angles are congruent.
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If these lines are parallel (let's say that they are parallel lines), then each pair of corresponding angles is congruent--only if the lines are parallel.
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If we don't have parallel lines--if I have lines like this and like this--they are not parallel;
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they don't look parallel, but I have a transversal--let's say 1 and 2: these angles are corresponding angles, but they are not congruent.
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They are not congruent, but they are still corresponding; angles 1 and 2 are corresponding angles, but they are not congruent.
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They are just called corresponding angles; so be very careful--only if the lines are parallel, then you can see that corresponding angles are congruent.
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They are the same; they have the same measure; they are congruent.
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Since these lines are parallel, I can say that angles 1 and 2 are congruent.
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So, angle 1 is congruent to angle 2; and it goes with all of the pairs of corresponding angles,
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like this one and this one--they are congruent...this one and this one, and this one and this one.
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Each of the pairs of corresponding angles is congruent only if the lines are parallel--that is very, very important.
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And that is a postulate; a postulate, remember, is any statement (such as this) that we can assume to be true.
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It doesn't have to be proved; if it is a theorem (the next few are actually going to be theorems),
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then they have to be proved in order for you to be able to use them, because it is not true until it is proven.
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The next one: here is a theorem; now, we are not going to prove these theorems now, but they are shown in your textbooks.
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The alternate interior angles theorem--just so you know, some kind of proof has to be shown for the theorems
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in order for them to be counted as true and correct, and then, that is when we can use them.
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But for now, since they are proven in your book, we are just going to go ahead and use them.
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The alternate interior angles theorem says that, if two parallel lines are cut by a transversal
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(meaning, if the two lines that are cut by a transversal are parallel), then each pair of alternate interior angles is congruent.
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Again, from the last lesson: if I have two lines...now, I know I am repeating myself a lot,
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but that is so that you will understand this, because I have seen a lot of students make careless mistakes with these,
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always thinking that these are congruent; in this case, if I tell you that these lines are not parallel,
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or if I don't even say anything about them being parallel, then you don't assume that they are parallel.
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We just have to assume that they are not parallel; then we can't say that angle 1 and angle 2 are congruent.
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We can say that they are alternate interior angles; that is the relationship; but they are not congruent in this case.
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So, for lines being parallel (now I am telling you that the lines are parallel), then alternate interior angles
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(let's say that this is angle 1 and angle 2)...angle 1 is congruent to angle 2.
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If the lines that are cut by a transversal are parallel, then alternate interior angles are congruent; and that is the theorem, the other one.
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The next one is the consecutive interior angles theorem: If two lines that are cut by a transversal are parallel,
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then (this is the tricky part--not tricky, but this is the part that students really make mistakes on) the consecutive interior angles
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are not congruent; they are supplementary--this is very important.
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Consecutive interior angles, we know, are angles that are on the same side, like these two angles right here.
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And they are both on the inside, the interior.
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So, angles 1 and 2 are consecutive interior angles; but then, they are not congruent--they are supplementary.
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Only if the lines are parallel, then consecutive interior angles are supplementary.
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See how the other ones that we just went over are congruent: these are not congruent--they are supplementary.
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You have to say that the measure of angle 1, plus the measure of angle 2, equals (supplementary means) 180.
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That means that this angle measure, plus this angle measure, equals 180--very important.
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And the next one: Alternate exterior angles, if the lines are parallel, are congruent.
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So, here is a pair of alternate exterior angles; angle 1 is congruent to angle 2.
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And that also works for this pair of alternate exterior angles, like 3 and 4; those will be congruent, also.
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Here we have parallel lines that are cut by a transversal.
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If AB (let's say that this is A, and here is point B--and these are the points, not the angles;
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here is point C and point D...then AB is a line, so it is line AB) is parallel to line CD,
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and line CA is parallel to line DB (and then I am going to add these parallel markers;
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that means that these two are parallel lines, and then for these--this is another pair of parallel lines,
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so that means that I have to draw two of them for these, because it is another pair), find the values of x and y.
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So then, here we have 80; and then I need to take a look at x.
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If I look for a relationship between this one and another one, even though these two have a relationship,
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this has a variable x, and this has a variable z.
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I would rather use this relationship, 4x and 80, because, if I am going to compare them, at least this one doesn't have another variable.
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So, it is easier to solve; so then, if I look at these two, I am only dealing with this line, line AB, line CD, and line BD.
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That means that line AC, I am going to ignore, because it is not involved in this pair of relationships.
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Remember from the last lesson: you look at the pair, and when you have the special pair, it only has three lines involved.
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It only has line AB, line CD, and line BD involved; the other lines that are there--cover them up.
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Those lines are there for another pair of relationships, so just cover it up.
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You don't need this line for this pair, so just ignore it.
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And then, to solve it, the theorem (and the relationship between these two: they are alternate interior angles,
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because BD would be the transversal between these two lines) says that if the two lines
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cut by a transversal are parallel (which they are--we know that because it gives us that in here),
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if the lines are parallel, then alternate interior angles are congruent.
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Since the lines are parallel, I can say that these two angles are congruent.
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Then, they are congruent, so 4x = 80; and I divide by 4: x = 20.
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There is my x-value; and then, for my y, let's look at this one.
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Well, with this one, I know that, since I have an 80 here, 80 is also congruent to this angle right here,
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because they are corresponding, and I know that these two lines are parallel.
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If these two lines are parallel, here is my transversal; that means that this angle right here and this angle right here are corresponding.
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And as long as the two lines that are cut by the transversal are parallel, then corresponding angles are congruent.
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So then, I can just write an 80 in here; and then, between this and this, they are vertical.
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Now, I could have just done this angle right here to this right here; so there are many ways to look at it.
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You can look at corresponding angles; if you didn't really see the alternate exterior angles--
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if that is kind of hard for you to see--then you can just say that, OK, they are corresponding, and then these two are vertical.
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And vertical angles, remember, are always congruent.
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So, you can say that these two are the same, because they are vertical.
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Or you can say that this and this are the same, because they are alternate exterior angles.
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And those are the same, as long as the two lines are parallel.
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So, either way: 4y + 10 = 80; then 4y = 70; so y = 35/2.
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And that is just 70/4, and then you just simplify it to 35/2.
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Now, it doesn't ask for the value of z, but let's just go ahead and solve it.
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We know that 6z and 80 have a relationship.
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Now, I know that this is 80, because we found x; x is 20; and 4 times x is 80;
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and also because they are alternate interior angles, so whatever this is, this has to have the same measure.
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So, either way we look at it: we can look at it as 6z with this one right here,
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or we can look at this one with this one right here--same relationship, same value,
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which also means that this one is also the same as 4x; this angle and this angle have the same measure.
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Either way, the 6z with this angle right here are consecutive interior angles, or same-side interior angles.
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Now, if the lines are parallel (which they are), then consecutive interior angles are supplementary--not congruent, but supplementary,
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which means that I can't make them equal to each other.
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Consecutive interior angles are the only ones that are not congruent from the special pairs of angles.
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Supplementary--that means that I have to make 6z + 80 equal to 180.
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6z = 100; z = 100/6, and then I can just simplify this to 50/3, and that is it; that is z.
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OK, the last theorem from this section in this lesson is the perpendicular transversal theorem.
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Perpendicular, we know, are two lines that intersect to form a right angle.
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So, if I have a line like this and a line like this, and they form a right angle, then they are perpendicular.
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But then, here we have a transversal involved; so in a plane, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other.
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Here are my parallel lines; I am going to show it by doing that.
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If my transversal, which is this line right here, is perpendicular to just one of the lines
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(it doesn't matter which one), as long as these lines are parallel (they have to be parallel),
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if it is perpendicular to one of the lines, then it has to be perpendicular to the other line.
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If this is perpendicular to this line, then it is going to be perpendicular to this line, as well.
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And that is the perpendicular transversal theorem.
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Now, if the two lines are not parallel (let's say like this), and then I tell you that this line is perpendicular to this line,
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it is not going to be perpendicular to this line, because these lines are not parallel.
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In this case, don't assume that it is perpendicular to both--that is only if the lines are parallel.
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Let's do a few examples: State the postulate or theorem that allows you conclude that angle 1 is congruent to angle 2.
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Now, remember: the only postulate was the corresponding angles one: that is the one where you have the angles in the same corner,
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in the same position, in the same corner of the intersection--that is the corresponding angles postulate.
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Everything else--the consecutive interior angles theorem, the alternate interior angles theorem,
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the alternate exterior angles theorem--those are all theorems; so the only one is the corresponding angles postulate.
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Here, what postulate or theorem allows you to conclude that angle 1 is congruent to angle 2?
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We know that this is our transversal line, because it is the one that cuts through two or more lines.
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Then, angle 1 and angle 2 are alternate exterior angles.
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Now, if these two lines are parallel, then we can conclude that angle 1 is congruent to angle 2; let me show that these two lines are parallel, too.
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Then, this would be the alternate exterior angles theorem.
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And this one right here--we know that these are corresponding angles.
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And the only way that the postulate will make them congruent (the only way we can apply the postulate) is if these two lines are parallel, which they are.
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So, I can say that, by the corresponding angles postulate, angle 1 is congruent to angle 2.
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All right, the next one: In the figure, line *e* is parallel to line *f*.
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So, let me show this; it doesn't matter which way--I can just do like this, or I can just do like that.
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AB is parallel to CD, so this one is parallel to this; and the measure of angle 1 is 73.
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I am going to write that in blue; so this is 73, right here.
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Find the measure of the numbered angles.
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All of the numbered angles is what it is asking for.
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Let's look at this: to look for the measure of angle 2, I know that angle 1 and angle 2 are supplementary, because they are a linear pair.
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They form a line, and a line is 180 degrees.
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So, all linear pairs are supplementary; so since linear pairs are supplementary, and these are a linear pair,
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I can say that 73 plus the measure of angle 2 equals 180.
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And then, to find the measure of angle 2, I have to subtract the 73; so the measure of angle 2 equals 107.
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And then, the next one: the measure of angle 3--well, if you look at this, we know that these two lines are parallel.
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This line intersects both of the parallel lines; so this is a transversal--this line segment AB is a transversal,
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which means that angle 2 and angle 3 are alternate interior angles.
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And by the alternate interior angles theorem, since the lines are parallel, we know that these angles are congruent.
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Since the measure of angle 2 is 107, I can say that the measure of angle 3 is 107.
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And then, the measure of angle 4: it is also alternate interior angles with angle 1, so by that theorem, again,
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since the two lines are parallel, those two will be the same; so it is 73.
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Then, the measure of angle 5 is corresponding with angle 5; angle 5 and angle 1 are corresponding,
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because it is as if I extend this line segment, just to help me out here: these two lines are parallel;
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here is my transversal; can you see that?--this is a line, and this is a line; here is that transversal;
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angle 1 and angle 5 are corresponding, so if this is 73, then the measure of angle 5 has to be 73.
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And then, the measure of angle 6--you can say that angle 6 is also corresponding with angle 3.
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So, if you extend this out again, there is my intersection, angle 3, and then my intersection, angle 6.
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The measure of angle 3 is 107, so the measure of angle 6 is also 107.
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Angle 7 is alternate interior angles with angle 6, so that has to be the same, since the lines are parallel.
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And the two lines involved would be this line and this line--can you see that?--this line and this line, and here is my transversal.
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These two lines are parallel, so angle 6 and angle 7 are congruent by the alternate interior angles theorem.
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And then, the last one, the measure of angle 8: it is supplementary with angle 6, because it is a linear pair.
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Or it is alternate interior angles with angle 5, or it is corresponding with angle 4; there are a lot of different relationships going on here.
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If you want to use the alternate interior angles theorem with angle 5 and angle 8, then it is going to be 73.
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If you want to look at the corresponding angles postulate with angle 4, then it is also 73.
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If you want to say that it is supplementary with angle 6 (it is a supplement to angle 6), then it is 180 - 107, which is 73.
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You can look at it in many different ways.
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That is it: see how all of the angle measures are either 73 or 107.
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Since all of these lines are parallel--these pairs are parallel, and those two pairs of lines are parallel--
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they are going to have only two different numbers, because all of their relationships are congruent or supplementary.
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So, it is either going to be congruent, or it is just going to be a supplement to it.
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Another example: BC is parallel to DE (that is already shown); the measure of angle 1 is 61 (this is 61);
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the measure of angle 2 is 43; and the measure of angle 3 is 35.
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This one is going to be a little bit more difficult, because we have lines that are closing in on the sides.
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And sometimes it is going to be a little confusing, or a little bit hard to see the lines that you need to see.
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And you are going to have to ignore these.
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So, look at angle...let's see...3 and angle 4; if you look at BE as a transversal, and these two
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as the lines that the transversal is intersecting, 3 and 4 are alternate interior angles.
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But these two lines are not parallel--those two lines that the transversal is intersecting are not parallel.
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So, you can't assume that they are congruent; you can't say that they are congruent, because look: the two lines are intersecting.
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Even though they are alternate interior angles, you can't apply the theorem saying that they are congruent, because the lines are not parallel.
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You have to be very careful; you can't say that angle 4 is 35 degrees.
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OK, so what can we say? We know that this line segment right here is parallel to this line right here.
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I can say that the measure of angle 5...because look at this: this angle 5 and angle 2 are alternate interior angles;
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now, let's see if we can apply the theorem and say that they are congruent.
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Here is my transversal; here are the two lines that the transversal is intersecting; are the two lines parallel?
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Yes, they are parallel; now, ignore this side and this side, AD and AE, because you don't need them.
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It is as if they are not even there; cover it up.
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Angle 5 doesn't involve those lines; angle 2 doesn't involve those lines.
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So, all you have to see is this right here; here is BC; there is angle 5 and angle 2.
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Here, these are parallel; here is 5, and here is 2.
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So then, these are alternate interior angles, and they are congruent, because their lines are parallel.
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The measure of angle 5 would be 43.
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And then, from here, I can say that the measure of angle 7...if you look at angle 7 and angle 1, I have a transversal;
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there is my angle 7, and there is my angle 1; these two lines are parallel.
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See how it is only involving the three lines, this line, this line, and this line.
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Ignore BE; see how I didn't draw it, because it is not involved.
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Ignore all of the other lines; just look at those three lines for angle 1 and angle 7.
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If it helps, you can draw it again; this one is a little bit hard to see using this diagram,
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so if it helps you like this, then just draw it again, just using those three lines.
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Angle 7 and angle 1 are corresponding; and since the lines are parallel, I can use the postulate to say that angle 1 and angle 7 are congruent.
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So, the measure of angle 7 is 61.
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So then, the ones that I found: this is 43; this is 61.
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OK, to find the measure of angle 4, I can say that, because all these three angles right here form a linear pair,
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that the measure of angle 7 plus the measure of angle 5 plus the measure of angle 4--they are all going to add up to 180,
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because they form a straight line; all three angles right here are going to form a 180-degree angle.
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You can say that the measure of angle...not 1....4, plus 61, plus 43, equals 180.
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The measure of angle 4, plus 104, equals 180; you subtract the 104, so the measure of angle 4 equals 76; here is 76.
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All of this is the one that I found, so I will write this in red: 76.
00:31:35.000 --> 00:31:55.200
And then, let's look at some other ones: now, if you look at angle 8, angle 8 also involves this parallel line.
00:31:55.200 --> 00:32:10.200
But this one is a little bit harder to see, because you have angle 8 like that; what is this angle right here?
00:32:10.200 --> 00:32:15.900
This is angles 2 and 3 together; it is this angle and this whole thing.
00:32:15.900 --> 00:32:26.200
Now, ignore this line; you are just involving this line, this transversal, and this bottom line DE.
00:32:26.200 --> 00:32:34.300
So, this BE is not there; so it would just be this whole angle together.
00:32:34.300 --> 00:32:41.400
So then, see how this angle right here and this angle right here are corresponding.
00:32:41.400 --> 00:32:49.700
But this has another line coming out of it like this to separate it into angles 2 and 3.
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All I have to do is add up angles 2 and 3, and that is going to be my angle 8.
00:32:55.600 --> 00:33:13.200
This is going to be 78 degrees, and since the lines are parallel, the corresponding postulate says that they are congruent; that equals 78.
00:33:13.200 --> 00:33:32.900
I will write that here: 78; and then, this 78 and angle 6 are going to form a linear pair.
00:33:32.900 --> 00:33:49.600
Right here, 78 +...now, since this is angle 6 right here, you can look at angle 6 and this angle right here,
00:33:49.600 --> 00:33:57.700
78 degrees, because that is angles 2 and 3 combined; they are going to be consecutive interior angles.
00:33:57.700 --> 00:34:08.900
And they are supplementary; so you can just do angle 6 + 78 = 180, which is the same thing as looking at this.
00:34:08.900 --> 00:34:15.500
These are supplementary, so angle 6 and angle 8 (78) are going to add up to 180.
00:34:15.500 --> 00:34:29.900
It is the same thing: the measure of angle 6, plus 78, is going to equal 180.
00:34:29.900 --> 00:34:48.000
The measure of angle 6: if I subtract 78, then you get 102, so this is 102.
00:34:48.000 --> 00:34:59.100
Now, with angle 9, to find the measure of angle 9, that is actually going to involve using the triangles,
00:34:59.100 --> 00:35:11.000
because the only relationship that this angle has with any of the other angles is that it forms within the triangle.
00:35:11.000 --> 00:35:20.100
And see how angle 9 is not supplementary; it doesn't form a linear pair; there is no transversal involved with angle 9.
00:35:20.100 --> 00:35:24.800
It is just these two angles, or those two right there.
00:35:24.800 --> 00:35:34.300
Angle 9 is actually going to involve what is called the triangle sum theorem, where the three angles of a triangle are going to add up to 180.
00:35:34.300 --> 00:35:46.400
So, we haven't gone over that yet; if you want, you can just say that the measure of angle 9, plus 61 (this angle),
00:35:46.400 --> 00:35:51.500
plus the 78, is going to equal 180, and then find the measure of angle 9 that way.
00:35:51.500 --> 00:36:03.500
You can also look at this big triangle and say that this angle, plus this angle, plus this angle, are going to add up to 180.
00:36:03.500 --> 00:36:15.100
You can also look at it as this triangle right here, saying the measure of angle 9 plus 75, together, and then 3, are going to be 180.
00:36:15.100 --> 00:36:19.600
And then, find the measure of angle 9 that way.
00:36:19.600 --> 00:36:26.600
So, for now, we are just going to solve for these; and that is it for this problem.
00:36:26.600 --> 00:36:31.900
The last example: Find the values of x, y, and z.
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Here you have three lines: now, these three lines are going to be parallel.
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I am going to make them parallel, so that I can solve for these values.
00:36:43.000 --> 00:36:48.900
Now, the only angle that is given is right here, 118.
00:36:48.900 --> 00:36:58.900
If you look at this, again, we have four lines involved; and to form special angle relationships, you only need three lines.
00:36:58.900 --> 00:37:08.300
You need the transversal and the two lines that it intersects to form those pairs of angles.
00:37:08.300 --> 00:37:17.900
Whichever lines you are using, always keep them in mind, and then look at what line you are not going to use,
00:37:17.900 --> 00:37:25.300
and ignore that line, since we have four and we only need three.
00:37:25.300 --> 00:37:42.100
Using this angle right here, 118, I can say that now this one right here and 11z + 8 are corresponding.
00:37:42.100 --> 00:37:53.500
And then, this one right here and this one right here are alternate exterior angles, because it is involving these three lines, and not this one right here.
00:37:53.500 --> 00:37:56.600
These would be alternate exterior angles.
00:37:56.600 --> 00:38:09.900
Or, if I ignore this middle line, and I just say that this transversal with this line and this line
00:38:09.900 --> 00:38:23.000
(again, ignoring the middle line--pretending it is not there), then 118, this angle right here, with x, would be alternate exterior angles.
00:38:23.000 --> 00:38:34.400
Imagine if you have a line, a line...here is your transversal; the middle line is not there; this is x, and then this is 118.
00:38:34.400 --> 00:38:38.700
You see that it is alternate exterior angles.
00:38:38.700 --> 00:38:44.700
So then, I can say that x is equal to 118, because the lines are parallel.
00:38:44.700 --> 00:38:57.300
And so then, I can apply the alternate exterior angles theorem, saying that that relationship, that pair, is congruent.
00:38:57.300 --> 00:39:05.900
The next one: let's look at z; this one right here, 11z + 8, is going to equal 118.
00:39:05.900 --> 00:39:18.500
Why?--because, if I look at this line, with this line and this transversal, they are going to be corresponding angles.
00:39:18.500 --> 00:39:26.200
And then, since the lines are parallel, the corresponding angles postulate says that they are congruent.
00:39:26.200 --> 00:39:46.800
11z + 8 = 118; so if you subtract the 8, 11z = 110; z = 10.
00:39:46.800 --> 00:39:51.900
There is my x; there is my z; and then, I have to find y now.
00:39:51.900 --> 00:40:07.100
For my y, I can say that this angle with 118--they are not congruent, remember, because they are going to form a linear pair.
00:40:07.100 --> 00:40:12.200
They form a line, so they are going to be supplementary.
00:40:12.200 --> 00:40:20.900
You can also note that this angle is 118, remember, because we said that they were corresponding--this one with this one.
00:40:20.900 --> 00:40:29.700
So, since this is 118, this angle with this angle would be consecutive interior angles.
00:40:29.700 --> 00:40:36.500
And if the lines are parallel, then the theorem says that they are supplementary, not congruent.
00:40:36.500 --> 00:40:56.100
So, either way, 3y + 2 =...not 180; you have to say that this whole thing, plus the 118, is going to equal 180.
00:40:56.100 --> 00:41:14.600
3y + 2 = 62, and then, if you subtract the 2, then 3y is going to equal 60; y is going to equal 20.
00:41:14.600 --> 00:41:22.000
x is 118; y is 20; and z is 10; just remember to keep looking for those relationships between the pairs.
00:41:22.000 --> 00:41:30.400
You can also definitely use the linear pair, if they are supplementary; you can definitely use that.
00:41:30.400 --> 00:41:38.700
If they are vertical, definitely use that, because you know that vertical angles are congruent.
00:41:38.700 --> 00:41:50.200
So, any of those things--you have a lot of different concepts that you learn that will help you solve these types of problems.
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That is it for this lesson; thank you for watching Educator.com.