High school geometry
- Proof: Opposite sides of a parallelogram
- Proof: Diagonals of a parallelogram
- Proof: Opposite angles of a parallelogram
- Proof: The diagonals of a kite are perpendicular
- Proof: Rhombus diagonals are perpendicular bisectors
- Proof: Rhombus area
- Prove parallelogram properties
Sal proves that opposite angles of a parallelogram are congruent. Created by Sal Khan.
What I want to do in this video is prove that the opposite angles of a parallelogram are congruent. So for example, we want to prove that CAB is congruent to BDC, so that that angle is equal to that angle, and that ABD, which is this angle, is congruent to DCA, which is this angle over here. And to do that, we just have to realize that we have some parallel lines, and we have some transversals. And the parallel lines and the transversals actually switch roles. So let's just continue these so it looks a little bit more like transversals intersecting parallel lines. And really, you could just pause it for yourself and try to prove it, because it really just comes out of alternate interior angles and corresponding angles of transversals intersecting parallel lines. So let's say that this angle right over here-- let me do it in a new color since I've already used that yellow. So let's start right here with angle BDC. And I'm just going to mark this up here. Angle BDC, right over here-- it is an alternate interior angle with this angle right over here. And actually, we could extend this point over here. I could call this point E, if I want. So I could say angle CDB is congruent to angle EBD by alternate interior angles. This is a transversal. These two lines are parallel. AB or AE is parallel to CD. Fair enough. Now, if we kind of change our thinking a little bit and instead, we now view BD and AC as the parallel lines and now view AB as the transversal, then we see that angle EBD is going to be congruent to angle BAC, because they are corresponding angles. So angle EBD is going to be congruent to angle BAC, or I could say CAB. They are corresponding angles. And so if this angle is congruent to that angle and that angle is congruent to that angle, then they are congruent to each other. So angle-- let me make sure I get this right-- CDB, or we could say BDC, is congruent to angle CAB. So we've proven this first part right over here. And then to prove that these two are congruent, we use the exact same logic. So first of all, we view this as a transversal. We view AC as a transversal of AB and CD. And let me go here and let me create another point here. Let me call this point F right over here. So we know that angle ACD is going to be congruent to angle FAC because they are alternate interior angles. And then we change our thinking a little bit. And we view AC and BD as the parallel lines and AB as a transversal. And then angle FAC is going to be congruent to angle ABD, because they're corresponding angles. Angle F to angle ABD, and they are corresponding angles. So in the first time, we viewed this as the transversal, AC as a transversal of AB and CD, which are parallel lines. Now AB is the transversal and BD and AC are the parallel lines. And obviously, if this is congruent to that, and that is congruent to that, then these two have to be congruent to each other. So we see that if we have opposite angles are congruent-- or if we have a parallelogram, then the opposite angles are going to be congruent.