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# Proof: Opposite angles of a parallelogram

CCSS.Math:

## Video transcript

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 C a b c a b is congruent to b DC so that that angle is equal to that angle and that a BD which is this angle is congruent to DC a which is this angle over here and to do that we just have to realize that we've have-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 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's make this let me do it in a new color since I've already used that yellow so let's say let's start right here with angle b d c-- well angle b DC and I'm just going to I'm just going to mark this up here angle B DC right over here it is an alternate interior angle with this angle right over here with that angle right over there and actually we could extend this point over here I could call this point I could call that point E if I want so I could say angle angle C D B is congruent to angle eb d angle e be d by alternate alternate interior angles this is a transversal these two lines are parallel a B or a E 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 a B as the transversal then we see that angle EB D is going to be congruent to angle BAC because they are corresponding angles they are corresponding angles so angle E BD e is going to be congruent to angle to angle BAC to angle BAC or I could say CA be angle c a.b they are corresponding corresponding angles and so if this angles convert to that angle of that angle is congruent to that angle then they are congruent to each other so angle C let me make sure I get this right c DB c DB or we could say BD c is congruent to angle C a B to C a B so we've proven this first part right over here and then to prove that these two are convert we use the exact same logic so first of all we view we view this as a transversal we view AC is a transversal of a B and C D and let me go here and let me create a let me create another point here let me call this point F right over here so we know that angle ACD angle ACD angle ACD is going to be congruent to angle FAC to FAC angle F AC because they are alternate interior alternate interior angles and then we change our thinking a little bit and we view AC and BD is the parallel lines and a B is a transversal and then angle FAC angle F AC is going to be congruent to angle abd because they're corresponding angles angle F to angle a BD and they are corresponding angles so the first time we view this is the transversal AC is a transversal of a B and C D which are parallel lines now a B is a 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 then or if we have a parallelogram then the opposite angles are going to be congruent