Definition of Corresponding Angles
Definition and properties of corresponding angles, showing the pairs of Try this Drag an orange dot at A or B. Notice that the two corresponding angles shown are the corresponding angles have no particular relationship to each other. Corresponding angles are formed when a transversal passes through two lines. The angles that are formed in the same position, in terms of the. Learn about parallel lines, transversals, and the angles they form. What's interesting here is thinking about the relationship between that angle right over there, and So if I assume that these two lines are parallel, and I have a transversal here, what I'm saying is They're always going to be equal, corresponding angles.
Well, I'll just call that line l. And this line that intersects both of these parallel lines, we call that a transversal. This is a transversal line. It is transversing both of these parallel lines. This is a transversal.
And what I want to think about is the angles that are formed, and how they relate to each other. The angles that are formed at the intersection between this transversal line and the two parallel lines. So we could, first of all, start off with this angle right over here. And we could call that angle-- well, if we made some labels here, that would be D, this point, and then something else.
But I'll just call it this angle right over here. We know that that's going to be equal to its vertical angles. So this angle is vertical with that one.
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So it's going to be equal to that angle right over there. We also know that this angle, right over here, is going to be equal to its vertical angle, or the angle that is opposite the intersection. So it's going to be equal to that. And sometimes you'll see it specified like this, where you'll see a double angle mark like that. Or sometimes you'll see someone write this to show that these two are equal and these two are equal right over here.
Now the other thing we know is we could do the exact same exercise up here, that these two are going to be equal to each other and these two are going to be equal to each other. They're all vertical angles. What's interesting here is thinking about the relationship between that angle right over there, and this angle right up over here. And if you just look at it, it is actually obvious what that relationship is-- that they are going to be the same exact angle, that if you put a protractor here and measured it, you would get the exact same measure up here.
And if I drew parallel lines-- maybe I'll draw it straight left and right, it might be a little bit more obvious. So if I assume that these two lines are parallel, and I have a transversal here, what I'm saying is that this angle is going to be the exact same measure as that angle there.
And to visualize that, just imagine tilting this line. And as you take different-- so it looks like it's the case over there.
Angles, parallel lines, & transversals
If you take the line like this and you look at it over here, it's clear that this is equal to this. And there's actually no proof for this. This is one of those things that a mathematician would say is intuitively obvious, that if you look at it, as you tilt this line, you would say that these angles are the same.
Or think about putting a protractor here to actually measure these angles. If you put a protractor here, you'd have one side of the angle at the zero degree, and the other side would specify that point.
And if you put the protractor over here, the exact same thing would happen. One side would be on this parallel line, and the other side would point at the exact same point. So given that, we know that not only is this side equivalent to this side, it is also equivalent to this side over here.
And that tells us that that's also equivalent to that side over there. So all of these things in green are equivalent. And by the same exact argument, this angle is going to have the same measure as this angle. And that's going to be the same as this angle, because they are opposite, or they're vertical angles. Now the important thing to realize is just what we've deduced here.
The vertical angles are equal and the corresponding angles at the same points of intersection are also equal. And so that's a new word that I'm introducing right over here. This angle and this angle are corresponding. They represent kind of the top right corner, in this example, of where we intersected. Here they represent still, I guess, the top or the top right corner of the intersection. This would be the top left corner. They're always going to be equal, corresponding angles.
However, the two other lines do not have to be parallel in order for a transversal to cross them, as you can see here: A straight angle, also called a flat angle, is formed by a straight line. The measure of this angle is degrees. A straight angle can also be formed by two or more angles that sum to degrees. Parallel lines are two lines on a two-dimensional plane that never meet or cross.
When a transversal passes through parallel lines, there are special properties about the angles that are formed that do not occur when the lines are not parallel. Notice the arrows on lines m and n towards the left. These arrows indicate that lines m and n are parallel. Corresponding angles are formed when a transversal passes through two lines.Applying Consecutive and Corresponding Angles to Prove Parallel Lines
The angles that are formed in the same position, in terms of the transversal, are corresponding angles. They are both above the parallel lines and to the right of the transversal. If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. A theorem is a proven statement or an accepted idea that has been shown to be true. The converse of this theorem, which is basically the opposite, is also a proven statement: If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.