G 9 M 12 L 7 1 Parallel Skew Lines and Transversals

The seventh lesson in module 12 is parallel line and transversals. We will look into the learning objectives. We we have to identify special angle pairs, parallel and skew lines, and transversals. Find values by applying theorems about parallel lines and transversals. Now, we will look into the keywords. Parallel lines, skew lines, parallel planes, transversal, complementary, interior angles, and exterior angles. Some of these terms you have already come across before, but some might be new like skew lines. So, we will look into them now. Parallel and skew lines. As you can see over here, there are two horizontal lines and one slant line. Parallel lines all can never intersect each other. Now, for example over here, JK and LM, the line JK and LM, you can see they are having the same slope but different intercept. They will never touch each other. So, these are parallel lines. They're coplanar lines that do not intersect. Whereas skew lines would have intersected, but they are in separate planes, so they do not intersect. The lines that do not intersect and they are in they are not coplanar. They're in separate planes, that's why. So, lines over here, you can see is in the plane and is in the plane and they are skew. Parallel planes, basically two planes which are parallel that do not touch each other or do not intersect each other are parallel planes. Over here, plane and Now, let us solve this problem over here. Identify parallel and skew relationships. Identify each of the following using the cube shown. The figure is given. Assume lines and planes that appear to be parallel or perpendicular are parallel and perpendicular, respectively. What they mean is look over here, this and this is supposed to be parallel, isn't it? If they look like that, yes, then assume it to be parallel. And now, for example, this and this is is perpendicular over here. So, then if it looks like perpendicular, then they are perpendicular. Now, they've given us the questions all lines skew to BC. So, this is the line over here, BC. Which are the skew lines which do not intersect it? Now, if you're thinking AD EF Wait minute because FD FE and ED are parallel lines, even GH. They're parallel lines, so don't consider them. Consider other than the parallel lines which do not intersect. Say, for example, AF, BE, even and GF. These are the skew lines. AF, BE, FG, and HE. So, just remember, don't consider the parallel lines, consider non-parallel but do not the ones which do not touch this line or do not intersect it. You can't consider AB or FB or any of these because they touch this line. Okay? So, that is about skew lines. And what about lines parallel to Where is is over here. What are the lines parallel to it? Parallel means same positions same slope but at different position. They'll not touch. So, now this is parallel line. Even this is parallel line and this is parallel line. So, AB over here, FG and CD. So, these are the three parallel lines. All all the lines parallel to so is this and these are the three parallel lines. And now, next one would be all planes parallel to the plane DCH, which is DCH. So, this is the plane. Now, which is the plane that is parallel to it? This plane ADEF? No, because it touches. Not even this plane. Not the up, not the bottom. Only one plane is parallel, that is ABG or AFG or AFB. This one. Okay? The one which have just now shaded, this one. So, the plane ABG is the plane that is parallel to DCH. So, that is about parallel and skew relationships. Now, we will quickly look into transversal as well. So, transversal is basically line which cuts at cuts two different lines at two points at different points. See now, line that intersects two or more lines in plane at different points is called transversal. You can see over here, these are the two lines and and would be the transversal. So, over here, there is transversal angle pair relationship. Now, this is very crucial because when transversal cuts two line, there are different angles that are formed and they are given separate names. Now, see let's see all the angles. Interior angles. Now, you can see this is the transversal. These are the two lines that are being intersected. You can see this will be These are the interior The angles which are inside where are where as this is exterior angles. So, interior angles would be the ones which are inside, 4 3 5 6, and exterior angles are 1 2 8 7. Now, consecutive interior angles mean which are not these are alternate, right? Sorry, adjacent. 4 and 3 are adjacent, not the adjacent. The opposite to it in the continuous manner. See, 4 and 5, 3 and 6, they are consecutive interior angles. Whereas alternate are the opposite ones, alternating. Opposite. 4 and 6 and 3 and 5. And then similarly, alternate exterior angles can be formed over here, that is 1 and 7, 2 and 8. And lastly, we have corresponding angles. This is See now, interior exterior is very simple and easy. Over here also, alternate interior exterior is very easy. Consecutive is also simple, continuous. The corresponding angles is very crucial. Corresponding means Now, imagine 1. 1 is at the top left side. So, in this particular line which from the transversal line you know, these four angles, it's top left. So, the corresponding angle to this will be the same top and left. So, top left would be 5. So, these both are corresponding. Now, if want corresponding angle of 7, that would be right down same way, right side down would be 3. So, 7 and 3. You can see 2 and 6, that means 2 6. So, this and this is corresponding. This and this. 6 and 2 and left down, 4. Left down is 8. So, those are the corresponding angles. So, with this information, we can easily solve few problems. Now, we have they have asked you to classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Now, over here, they have given us few angles. So, 4 and 5. Let's see what is this. When you look at it, you must understand they are inside, interior, and opposite direction. So, alternate interior angles. Now, what about the next one? 3 and 7. 3 7. See, this is at left down. This is also left down, so they are corresponding angles. Now, 3 and 5. 3 is over here, 5 is over here. They are in the same interior and consecutive, so they are consecutive interior angles. 1 and 8. Let's see where is 1 and 8. 1 is over here, 8 is over here. Alternate, but they are outside, so alternate exterior angles. So, please try to understand. Interior means inside, exterior means outside, alternate means opposite like this. Consecutive is same and corresponding is the one which is relating to the position. If it is top left, this also must be top left. So, let's again identify transversals and classify the angle pairs. Identify the transversal connecting each pair of angles in this photo. Then classify the relationship between each pair of angles. So, now over here, see there are how many transversals? So, now these are all the angles formed. They are not given all. Let's consider the first transversal We can see 1 3 4 1 2 3 and 4. You You consider any and you can write the relation. Now we can consider two and three because the relationship between these both is very easy to know, that is corresponding angles about two and three. Because now we do not have proper relation between two and four. It's not interior, it's exterior. So you can find the this and this that's exterior, but still anyone is enough over here. Two, three, you know the relationship is corresponding angle. If at all this angle was some angle was over here, then it would be alternate interior. Okay? So now then let's take another example another line that is that's transversal now. And over here they have considered seven and six. You can consider any, but over here you can see seven and six is known. So these are consecutive and interior, consecutive interior angle. So next would be transversal and you can see which are known. So over here there are all these angles known. So it's up to you to choose. You can say you can take seven and one, they're external. Over here now one and eight is taken because they're corresponding angle or you could have taken even seven and one, they're external alternate angles. It's up to you. So it depends on the options basically. And over here transversal over here they have taken six and three. This is alternate exterior angles. Sorry, told external, it's exterior angles. So now over here now for example in the transversal only two and three is mentioned, but you can also consider one and four that is alternate exterior angles. It's up to you, it depends. At least one relationship is enough over here, that's why only one is considered, but you can take any, it's up to you. Even one and four is correct if you have written alternate exterior. So mainly if this comes in MCQs, so look into the options thoroughly and see for the relationship between them.