Angles between intersecting lines
Identify all sets of parallel and perpendicular lines in the image below. So let's start with the parallel lines. And just as a reminder, two lines are parallel if they're in the same plane, and all of these lines are clearly in the same plane. They're in the plane of the screen you're viewing right now. But they are two lines that are in the same plane that never intersect. And one way to verify, because you can sometimes-- it looks like two lines won't intersect, but you can't just always assume based on how it looks. You really have to have some information given in the diagram or the problem that tells you that they are definitely parallel, that they're definitely never going to intersect. And one of those pieces of information which they give right over here is that they show that line ST and line UV, they both intersect line CD at the exact same angle, at this angle right here. And in particular, it's at a right angle. And if you have two lines that intersect a third line at the same angle-- so these are actually called corresponding angles and they're the same-- if you have two of these corresponding angles the same, then these two lines are parallel. So line ST is parallel to line UV. And we can write it like this. Line ST, we put the arrows on each end of that top bar to say that this is a line, not just a line segment. Line ST is parallel to line UV. And I think that's the only set of parallel lines in this diagram. Yep. Now let's think about perpendicular lines. Perpendicular lines are lines that intersect at a 90-degree angle. So, for example, line ST is perpendicular to line CD. So line ST is perpendicular to line CD. And we know that they intersect at a right angle or at a 90-degree angle because they gave us this little box here which literally means that the measure of this angle is 90 degrees. By the exact same argument, line the UV is perpendicular to CD. Let me make sure I specified these as lines. Line UV is perpendicular to CD. So I did UV, ST, they're perpendicular to CD. And then after that, the only other information where they definitely tell us that two lines are intersecting at right angles are line AB and WX. So AB is definitely perpendicular to WX, line WX. And I think we are done. And one thing to think about, AB and CD, well, they don't even intersect in this diagram. So you can't make any comment about perpendicular, but they're definitely not parallel. You could even imagine that it looks like they're about to intersect. And they give us no information that they intersect the same lines at the same angle. So if somehow they told us that this is a right angle, even though it doesn't look anything like a right angle, then we would have to suspend our judgment based on how it actually looks and say, oh, I guess maybe those things are perpendicular, or maybe these two things are parallel. But they didn't tell us that. And that would actually be bizarre because it looks so not parallel. And actually then this would end up being parallel to other things as well if that was done. It's a good thing that wasn't because it would look very strange. But based on the information they gave us, these are the parallel and the perpendicular lines.