 # 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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Okay. Hello. We're back with more on parallel lines. Recall if you have two lines that are parallel one way of indicating that, by the way, would be these arrows that point in the same direction. Or I could say, And what I did not mention last video is if this is lying l and this is line em. I can write it as l is Parallel thio. M You basically have these two, um, tilted lines that represent parallel lines. Remember that this line right here is what we call a trans Verceles. And it's the Trans Verceles interacting with the two parallel lines that create eight angles. And remember, we have lots of different types of angles we have which have nothing to do with the parallel lines is we have vertical angles, those angles that are basically created when two lines cross each other where the X is formed, you also have corresponding angles. So, for example, angle one an angle five, which are in the same relative position. They're gonna be congruent. You also have angles like angle three and angle six, which recall alternate interior. They're gonna be congruent and for that matter, angles eight and angles one or alternate exterior, they're gonna be growing. But then you have another set of angles that are not going to each other, but more so, uh, supplementary to each other. And that would be any angles that are on the same side either interior exterior. So, for example, these two angles are going to be supplementary, which means they equal. They have some of 180 degrees. Likewise, this angle and this angle also are supplementary, which remember, that means that equal 1 80. So this is kind of what we're dealing with here. And if if you have parallel lines, you have all of the angle relationships. And if you have these annual relationships, you also have parallel Allianz's A by conditional statement if parallel than angle relationships, if angle relationships, then parallel. So let's work on a few of these problems here. Okay, so we're looking at this worksheet and based upon what we know about parallel lines, what you notice all those little red arrows are indicating that we have parallel lines. Doesn't matter if you have one or two it just make sure that they match in terms of the number of heroes. There's no reason to have to There. If you had a diagram, it was more complicated. Had multiple lines like multiple pairs. Then you're gonna wanna use 123 and so forth triangles to indicate which lines are parallel to each other. So in question one, you know, what's the what's the value of the indicated angle? Well, this is gonna be 75. This has nothing to do with parallel alliance. It just happens to be that these air vertical angles because they're formed by the, uh, intersection of two lines. So you have vertical angles, their vertical angles there. Um, for question three. Remember these air these two angles air, same site interior, so they're gonna have a sum of 1 80. So if you take 55 away from 1 80 you're left with 1 25 for your answer here. Yeah. Question 60. I didn't mention this because this has nothing to do with parallel lines either. But remember, from a couple of lessons ago, if you have an angle that's exactly a straight line, it's 100 80 degrees. So if I draw any kind of ray, then I create two angles that are supplementary because the equal 1 80. So I'm gonna take 1 80 minus 68. I'm going to get 112 here for this angle because 1, 12 and 68 equal 1 80. Um, if you recall from our previous lesson this angle and the one we're looking at our alternate interior, anything that has the alternate in front of it indicates congruence e. So we have another angle. That's 113 degrees. Question five if you recall this angle and the angle in the same relative position. So in reference to the Trans Mursal, those two angles are both upper left their congruent as well. You get 89 degrees on question six. Notice that this angle is alternate exterior from that meaning they're on opposite sides of the Trans Mursal outside of the parallel lines. Again, alternate exterior angles are equal, so I could go ahead to say 90 degrees. Okay, on question seven, you have a right angle. So that's the symbol for perpendicular. This is also a fancy form of talking about perpendicular upside down T things going to indicate that we have 90 degrees. It should make sense then that all of the angles are gonna be 90 degrees because 1990 makes 1 80. So ultimately, we have 90 degrees everywhere. What type of angles? With ease. Well, looks like we have alternate interior. That's congruent 130 degrees. What kind of angles or question? Nine. Well, two lines intersecting their vertical vertical angles are automatically equal 1. 30 again. Similar Question eight. Alternate interior, 190 degrees. Okay, When it comes to geometry, it is very easy to bring in algebraic concepts. And so we're gonna take what we know about the parallel lines and the angle relationships that are formed. And then we're gonna use algebra, um, to basically solve the problem. So on question 11, these angles here are in the same relative position, their corresponding angles. So what we know about corresponding, they're congruent so I can set them equal to each other. Bring the Nynex over, bring the five over, we get six equals X. Be careful. If the question then asked what is theatrical measure? You would plug six back into either here or here and let's say, 10 times six plus five, you're going to get 65 so each of those angles as a value of 65 degrees. Okay, so what's true about let's say these angles? Well, they're both exterior and on opposite sides. So therefore, they're alternating anything that's alternating Arkan growing so automatically have 80 equals 20 x or, in other words, four equals. X on question 12. Just kind of skipping around here. Here you have opposite sided angles from the Trans Verceles inside. So their alternate interior Remember anything with e alternate in front of it. We have congruence e. So 60 is gonna equal negative five plus 13 X or in other words, 65 equals 13 x and 65 divided by 13 gets us a total of five and you get the idea. So you're essentially going to either be setting two angles equal to each other in the situation of corresponding vertical or anything that's alternate interior, alternate exterior, anything that same side, interior, same side exterior. Let's say 14, these air supplementary. So instead of setting them equal to each other, you're gonna say 69 plus 12 x plus 1 15 equals. Pardon me 1 80. So that gets you and see if we can combine things. On the one side we get 69 1 people 1 84 plus 12 X equals. So if I subtract 1 84 from 12, you actually get and this should be a to B 12 export me. You're actually going to get 12. X equals negative four. Or, in other words, X equals negative. One third. I think we have an issue here. 69 plus 12 x plus 15. You know what I did? I put 115 here to see how I had a hoops to see how I knew I had an issue. And that's because, well, everything got erased. That's alright. E knew I had an issue because when I was doing the work here 12 again, I got it raced when I was doing the work. I gotta get negative fraction. I don't care what the fraction? It's okay. I don't even care. It's negative, but it wasn't making sense. Let me see here. That's why we're off off by a page. Thank you. This negative one third doesn't make any sense because I wanna plug this back into here. There's no way I'm going to get a 69 So I put 1 15 when I should have put 15. So let me try that again. So there should be some kind of gut feeling that says you're doing the right thing again. First of all, you've got one of two conditions. Either they're equal or their supplementary. So let me try this again. 69 plus 12 X plus 15. Not 1 15 equals 180 69 plus 15 is 84. You have 12 X plus 84 equals 1 80. I'm gonna subtract 84 and I'm gonna get 96. We have 12 X equals 96. And like most of these worksheets, they're meant to have nice numbers. You get X equals eight. And again, if they were asking for, you know, the value of the angle, you could plug that back in, Although we automatically know in 69 So here's what's going on. You either have, like, supplementary angles, so you're gonna add them together, send people to 1 80 or there's something like vertical angles where you're going to set them equal to each other. Um, so you have all these angle relationships now you have to identify what they are. Is it a congruent thing where you set up with each other? Is that a supplementary thing? Where you Adam together instead of 1 80? But essentially now here's the algebra overlapping the geometry and several examples of how you can use parallel lines to solve forgiven value or determine different angle measurements, Thanks. Webster University
##### Top Geometry Educators ##### Kristen K.

University of Michigan - Ann Arbor ##### Samuel H.

University of Nottingham 