 # In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. Parallel planes are planes in the same three-dimensional space that never meet.

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Hi, Mr. Coming back with you, we're gonna talk about parallel lines today. You might have heard about a gentleman called you Could. He lived about 2000 years ago roughly and is known as the father of geometry. He is known for basically compiling all of the geometric knowledge and other topics, but mostly geometric knowledge of the time. And he put it together in a very organized and logical progression. And this set of books called The Elements at the beginning. He basically has five postulates and a and a possible to something that we all accept to be true. We don't need proof. For example, a line Is that a set of points that go in opposite directions forever? There's no proof of that. We all just kind of accept what the line is. But the fifth partial it was a little bit more complicated than that. Stated this if you have two lines and a third line that crosses those two lines, we called this line by the way, a Trans Verceles. Okay, if these two angles okay, are less than 180 degrees. So if these two angles added together are less than 1 80. Then eventually these two lines you're going to cross at some point on this side of the Trans Verceles. Same thing goes for the other side of the Trans Verceles. Now, no one really argued with that. It just didn't seem to be kind of a dust statement. People wanted to prove it. And so it's like 2000 years. They tried to prove it, and they found out that you couldn't prove it. And that's a story for a whole other day. One of the story. However, if we're living on a flat surface, if you're living on a plane, this is a true statement. Okay, I'm gonna write that over again. If you have two lines crossed by a third and these two angles here add up to 1 80 then the lines are parallel. And another way of saying that is if we know the lines of parallel. Those two lines are those two angles. Pardon me, add up to 1. 80. So these two added together equal 1. 80 and these two added together equal 1 80. Okay, so parallel lines are essentially lines that lie in the same plane that air forever. Equi distant from each other. Okay, the distance between the two lines will never change that distance. D will always be consistent, will never increase or decrease. It will always remain constant. That's what we call parallel lines. Now again, The key word there was They lie in a plane. So the co plainer you could have two lines that air in different planes. Those airlines air called skew. So we're not talking about skew lines of talking about co player lines that are forever equi distant from each other. That's what we call parallel lines. Some very, very special angle relationships are created when we deal with parallel lines. So essentially we have eight angles that air created when you have parallel lines. And they're basically two major relationships between all of them. So first we have angles. One angle, four angle five and angle ate all of these air equal to each other. They're all couldn't grow it. That's the symbol for congruent. Okay, so why are they equal? Well, a couple things. One is angle one and angle four what we call vertical angles. Anytime you have two lines cross, you have two pairs of vertical angles. Okay, basically the angles that formed the X on either side of the X that were called vertical angles. They're equal to each other. So so, for example, one and four vertical, three and two or vertical five and eight, a vertical six and seven a vertical. But the reason why 145 and eight equals because angle one an angle five are called corresponding angles. Because relative to the trans Verceles angle, one angle five are in the same relative position. They're both in the upper left hand spot, just like two and six are both in the upper right hand spot three and 7 40. Those air, known as corresponding angles corresponding angles are equal. So if you think about it, if one in four, if one and four are equal because their vertical and one in five or equal because their corresponding that makes 145 and eight all equal, so you have that relationship. Likewise, I could say angles to 36 and seven are also equal, so lots of things fall from this. If you accept that corresponding angles are equal, we can then come up with a whole slew of relationships. For example, one in five or equal to and six or equal three and seven or equal in four and eight are equal all because their corresponding angles. But now, if I do some substitution, Aiken form different groups of equal angles. So, for example, I could tell you that angle three and angle six or equal because of two in six are equal and two and three equal because the vertical that forces three and six to be equal and I did mention that already. But they have special names. Angles three and angles six, along with Angles four and angles five. They're known as alternate interior angles. Alternate because they are on alternating sides of the Trans Verceles and they're in the interior of the parallel of the parallel alliance. So three and six air known as alternate interior four and five are also known as alternate interior. So I draw that picture over again. Okay, so just like three and six were alternate interior four or five or alternate theory by that same logic, one in eight as well as angles, too. And angles seven are known as alternate exterior angles, and they're also congruent toe one another. Okay, Now, We also have one other set, our family of angle relationships. And that is, do this a different color. Angles three and five, which are on the same side of the trans trans murschel. They're actually supplementary. This is the word we haven't used yet. Supplementary means two angles that add up to 180 degrees, So angle three plus angle. Five Equal 1 80. Likewise angle four plus angle. Six. Equal 1 80. They're known as same side interior, and they add up to 1 80 similar fashion one in seven and two and eight. They're known as same side exterior. So just like we have alternate interior alternate exterior angles, which are congruent to each other, we have these set of angles called same side interior or same side exterior angles that are supplementary or, in other words, at 2080. So given that you have two parallel lines and the trans Verceles the civil, by the way to show that the parallel these two arrows that air each on top of the respective parallel lines pointing in the same direction that would tell you that the parallel that we have to say that they're parallel Once we have parallel lines, we have these eight angles. You call it anything you want. But we have these eight angles here and we have equal angles, which I'll do in red. So one in 45 and eight are all equal and we also have. I'll do this in yellow to three, six and seven are also the other family of equal angles. But then you have the same side interior which are supplementary just like three and five are. But just like one in seven or same side exterior there, supplementary. So you have this whole family or these groups of equal angles. You have this whole groups of supplementary angles because of the parallel lines. Now, if you talk about if then statements if you have parallel lines, you have all these angle relationships. We're also going to find out if you have all these angle relationships thing, you are guaranteed parallel lines. So, for example, if I told you, I have these two lines crossed by a third and I have angle a an angle B and I told you, angle a equals or is congruent angle B. I would say, Hey, that means that this angle and this angle are alternate interior angles are equal to each other. That would force these two lines to be paralleled, So it's actually a by conditional statement. If lines are parallel, all these angle relationships exist. If he's angle, relationships exist, even one of them, then the lines are parallel. And then all of the angle relationships exist. So that's, uh, basically a quick run down of parallel lines. Again, Uh, it comes from you. Clips, fifth postulate. But essentially in our world, parallel lines are two lines in the same plane that air forever equi distant from each other. Thanks. Webster University
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