 Here's a quick geojabrit demonstration to show you if you have two lines cut by a transversal So that a pair of corresponding angles are congruent then those two lines are parallel a Quicker way of writing that would be if you have corresponding angles that are congruent then the lines that created them are parallel So Here I have an angle and I can change the value of the angle and if I create a Corresponding angle now first of all what is a corresponding angle a corresponding angle is an angle that is in kind of the same position But on a different line in other words m and n are these two lines cut by a transversal t and Corresponding angles are angles in the same spot now I could say this angle angle a b c is kind of in like the northeast Quadrant if we wanted to think about it that way So northeast quadrant and northeast quadrant and I see that these two lines Regardless of what the angle is those two lines will be parallel So this is a construction of Euclid and if you wanted to learn a little bit more about that Feel free to ask your teacher or Google around But basically the point is if we have corresponding congruent angles then lines are parallel and now we can use that statement in further proofs Take a minute to read this if then statement the if part refers to given information In other words, we're given that we have two lines cut by a transversal and alternate interior angles are congruent The conclusion of the if then statement is what it is. We're trying to prove we want to prove that lines are parallel So let's draw out the given information So here we have two lines cut by a transversal let's call one of the lines m and We could call the transversal t Now our job is to show that alternate interior angles are congruent But which are the alternate interior angles? So the alternate interior angles could refer to let's say this angle and This angle those two blue angles it could also refer to these two green angles Now for sake of argument, we only need to pick one of the angle pairs. So I think I'll pick the blue angles So we're given that those two angles are congruent It'd be easier if I could just name the angles. Let's call this angle one and angle two So we know that angles one and angle two are congruent Now as it stands right now angles one and angle two are congruent The only way that we can prove lines are parallel are with corresponding angles. So corresponding angles Take a look at angle one angle one and its vertical angle. Let's call this angle three. I Know that angles one and three are congruent because vertical angles are always congruent So now we know angles one and three are congruent because one and three are vertical angles So if one is congruent to two and One and three are also congruent, then we can say that angles two and three are also congruent So two and three form corresponding angles and corresponding angles will allow us to say that the lines are indeed parallel. So that's kind of the general idea of the proof. Now let's put it together in a rigorous proof. So let's restate the given and prove information in terms of our diagram. So we're given one and two are congruent. In other words, alternate interior angles are congruent, and our job is to prove that m and n are parallel. Angles one and angle two are congruent, and we know that's true because it's given information. That's the hypothesis of the conditional statement. Also we know that what were they angles one and three? Yeah, angles one and three are vertical angles. So we'll want to mention that fact and then state that vertical angles are always congruent. So angles one and two, that's a typo, not angles one and two. Angles one and three are vertical angles, which means angles one and three are congruent. Angles one and three are vertical angles, and therefore vertical angles are congruent. Now one and two are congruent, as shown in the blue. One and three are congruent, as shown in the green. So the blues and the greens can come together. If one is two and one is three, then two and three are congruent. So angle two is congruent to angle three. So two and three are congruent by the transitive property, right? We can bring together the blue and the green, and if two and three are congruent, two and three are corresponding angles, and corresponding congruent angles always make parallel lines. So that last box we can fill in would be that m and n are parallel. And why is that true? It's true because if we have corresponding angles that are congruent, then the lines are parallel.