Whether a special quadrilateral can exist

An interesting question has come from some of the engineers here at the Khan Academy while they were working on the code. Are there any quadrilaterals-- And I've drawn an arbitrary quadrilateral right over here. Quadrilateral ABCD. But is there any quadrilateral that if I were to draw the diagonals, so let me draw the diagonals. So one diagonal is BD. And then the other diagonal, I'll draw it right here, is AC. And the point of intersection of these two diagonals is, let's call that point E. Is there any quadrilateral where angle AEB is congruent to angle ECB is congruent to this angle right over here? ECB. So I'll let you think about that for a second. Is there any quadrilateral where these two angles are going to be congruent to each other? Now to think about whether this is possible, let's assume that it is. So let's assume that we do have a quadrilateral where this is indeed the case. Where AEB, this angle right over here, is congruent to angle ECB. We're just going to assume that right from the get-go. Now let's try to visualize this whole thing a little bit differently. DB is a segment, but it's a segment of a larger line. So we could keep extending it off like this. And let's call that larger line, line l. Let me draw it like this. So this right over here is line l. DB is a subset of line l. And CB, which is a segment, is also a subset of a line. And we could call the line, if we were to keep extending it, let's call that line m. So I'll draw it like this. So line m. CB is a subset of line m. And then we also have the segment AC. And AC, once again, is a segment but we can view it as a subset of a larger line. And we'll call that larger line, line n. And so let me draw a line n like this. And we see that it intersects both line l and line m. So let me draw it like this, so it looks something like that. That is line n. And we've assumed that angle AEB is congruent to angle ECB. Well point E is just where n and l intersect. So this right over here is point E. And point C is where lines n and m intersect. So this is point C. So this assumption right over here, that we're assuming from the get-go, is saying that this angle AEB-- I'll do it in the same magenta-- AEB is congruent to this angle. ECB is congruent to that angle. These angles are these angles right over here. Now what does that tell us about lines m and l? Well the way we've set it up, we have two lines, lines l and m. Line n is a transversal. And now we have two corresponding angles are congruent. We assumed that from the get-go that we could find two quadrilateral, where these two corresponding angles are congruent. But if you have two corresponding angles congruent like this, that means that these two lines must be parallel. So this tells us that line l is parallel to line m. Line l is parallel to line m. Which means that they will never intersect. But we have a contradiction showing up. And I'll let you think about what that is for a second. Well the contradiction that shows up, is that if line l is parallel to line m, then any subset of those two lines have to be parallel to each other. So if l is parallel to m, then that tells us that DB, segment DB, needs to be parallel to segment CB, which means that they can never intersect with each other. But that's a contradiction. They do intersect with each other. They intersect right over here. So if we assume that there's a quadrilateral, where this is the case, you end up with this contradiction, something that's not possible. By definition it was a quadrilateral. This diagonal intersects at point B, which is where this side intersects, too. And so they have to intersect. These two cannot be parallel in order for this to actually even be a quadrilateral. And because of that, it is impossible for any quadrilateral for this angle AEB to be congruent with this angle ECB.