 Here's a quick demonstration to show you that if you have two parallel lines cut by a transversal then each pair of corresponding angles are congruent So here in GeoGebra, I have two parallel lines. I constructed them to be parallel So I know that they'll always be parallel and now we need So in the hypothesis we have two parallel lines Cut by a transversal. So let's include a transversal now I have a transversal it's AC and and Again the statement says if I have two parallel lines cut by a transversal, which I now have The conclusion is each pair of corresponding angles are congruent. So corresponding angles The way I think about corresponding angles is I think about a compass Sort of a north-south compass a Corresponding angle is one that in on one line is in the same position as it is on the other line An example of corresponding angles would be these two red angles They're corresponding because they're sort of both in the northeast quadrant of this This intersection and this intersection so Each pair of corresponding angles are congruent We can see right now both angles are 112.2 degrees and if we move the parallel lines Those angles stay fixed. They're always going to be congruent to each other Now furthermore each pair of corresponding angles are congruent So the pairs of corresponding angles refer to these angles. I see that the two greens Let's say this angle and that angle those are congruent because they're corresponding Likewise, these two reds down below Corresponding so all of these angles are congruent because we have parallel lines So once again, if we have two parallel lines cut by a transversal Then each pair of corresponding angles are congruent and we can abbreviate that with if parallel lines Then corresponding angles are congruent. Take a minute to read this statement. We're about to prove The statement highlighted in yellow is the hypothesis of the conditional statement The hypothesis is the given information two parallel lines cut by a transversal The conclusion is that which we're trying to prove We want to prove that the alternate interior angle pairs are congruent So first let's draw out the given information We have two parallel lines cut by a transversal. Let's show that they're parallel and give them names. I Called them lines m and n cut by a transversal t and remember the way to show lines are parallel Now our job is to show that each pair of alternate interior angles is congruent If we show that one pair of alternate interior angles is congruent Then the same logic that we use to show that that pair is congruent will apply to the other pair So let's pick a pair of alternate interior angles. Let's call them one and two So our goal is to prove that angles one and two are congruent So in terms of given information, we're given that m and n are parallel and our job is to prove that angles one and two are congruent So first off, let's deal with our given information Parallel lines remember parallel lines will always deal with angles in some way and in the previous video We saw that parallel lines make congruent So those corresponding angles let's call the green ones angle three and then angle two was originally named So now we have angles two and three are congruent. So once again because we know m and n are parallel We can say that Angles two and three are congruent because parallel lines make corresponding congruent angles Now we've established that two and three are congruent. Let's take a closer look at angles one and three Now vertical angles, vertical angles aren't necessarily anything that's from the given information But it can be implied from the given information because whenever lines intersect they form vertical angles And we know that vertical angles are always congruent So we've established that angles one and three are congruent So one and three are congruent two and three are congruent So by the transitive property if one is congruent to three and Two is congruent to three Then we can say that one and two are congruent to each other and now we've established that angles one and two are congruent which is what we were set out to prove. Angles one and two are congruent, and so therefore we've proven the statement if two lines, excuse me, if two parallel lines are cut by a transverse then each pair of alternate interior angles is congruent. That's a great proof! Nice job!