 Hi, my name is Adam. I'm a student of Dr. Missildine's at SUU and today we're going to be exploring and proving the alternate interior angle theorem. So the alternate interior angle theorem says that if two distinct lines cut by a transversal have a pair of congruent alternate interior angles, the two lines are parallel. This is a very, very big theorem in congruent geometry. So let's start by constructing what the scenario says. So I'll have two lines cut by a transversal. Let's call the lines L, M, and G. Alright, and let's label six points on here. I have a point here, B, A, B. Call this one A prime, and this one B prime. Alright, and by assumption in the theorem, we assume that angle APQ is congruent to angle B prime QP, because that's what it means to be alternate interior angles. So let's go ahead and show that these two angles are congruent. Alright, so to prove the alternate interior angle theorem, we're going to go ahead and do it by contradiction. So to contradict this, let's go ahead and assume that these lines L and M intersect, and we're going to say they intersect at some point R. And we're going to go ahead and put R over here. And without loss of generality, we're going to say that R is on the same side of T as A and B are. Alright, so what can we say now? Well, by segment translation, let's translate this segment PR here to Q on the ray QB prime. Okay, so there's going to be some point up here. Let's call it R prime such that PR and QR prime the segments are congruent. Okay, and then we'll know that the triangles are congruent because this is congruent to itself. These two angles are congruent, and this is congruent to this by the segment translation we just did. Okay, so triangle PRQ is congruent to triangle PQR prime. So for now, we're going to go ahead and draw a line like this and say that this triangle here is congruent to this triangle here. Alright, since these triangles are congruent as corresponding parts from different triangles are congruent, we know that this angle R prime PQ is congruent to this angle because those are the corresponding parts of the congruent triangles. This angle is angle PQR. Alright, additionally, since angle APQ and B prime QP are congruent and A prime PQ is the supplement of APQ and BQP is the supplement of B prime QP since supplements of congruent angles are congruent. Angle BQP is congruent to angle A prime PQ. So therefore, by transitivity, we know that angle A prime PQ is congruent to angle R prime PQ. However, by uniqueness of angle translation, there can only be one angle here. This smaller angle can't be congruent to this bigger angle unless they're the same angle. Therefore, it's too big, this line goes through A prime because it has to be congruent. So this should look more like this. Alright, this shows that R prime is an element of the ray PA prime and more importantly it shows that R prime is an element of the line L and it's an intersection of L and M because we already know that R prime is on the line M. However, what that gives us is that both R over here and R prime are on or are elements of the intersection of L and M and we know that R is not the same as R prime because there are opposite sides of T. We have that plane separation in congruence geometry. So lines L and M have two different points of intersection R and R prime which violates line determination. So this situation can't happen which contradicts our original assumption that these lines intersect and the contradiction of these lines intersecting is that the lines are parallel. Therefore, if two distinct lines cut by a transversal have a pair of congruent alternate interior angles, the two lines are parallel and that is the alternate interior angle theorem. Thank you for watching.