 We're asked to write a flow proof for the following parallelogram theorem. We have the if statement. If a quadrilateral is a parallelogram, that's always going to be our given math, parallelogram math will be our given statement. The then statement, its opposite angles are congruent, is going to be what we try to prove. Our opposite angles, angle M and angle T, are going to be what we're trying to prove. There's a lot of different ways to approach flow proofs. I always like to start by putting the given information on our illustration and then deciding what postulator theorem we're going to use. And that's why you probably want to have out that DPT postulate theorem sheet to help us as we go along here. Anytime we're working with parallelograms, we know that opposite sides are parallel. So you can go ahead and do the markups to show that we have two sets of parallel sides. That should tell you whenever you're working with parallel lines, we're going to have some angle relationships and we have two sets of parallel lines. So we're going to have two sets of angle relationships. We know that if AT and MH are parallel with our transversal right here, then we have a set of alternate interior angles that are going to be congruent. So I know that angle 2 is going to be congruent to angle 4 based on those two opposite sides being parallel. And at the same time then with side MA being parallel to TH, that gives us the other set of alternate interior angles. So I have two sets of alternate interior angles. The third piece that's going to finish this to give us our theorem of why these two triangles can be congruent is the shared side. We know that AH is congruent to itself. So why don't you go ahead and mark that up? This is a nicer version, more readable. And I always like also to label the As and the S's so that we make sure we get the correct theorem to put into our flow proof. Once we have that mark with the angle side angle, we can go ahead and start there. I like to start in the middle and put my three boxes there and even label the ASA so I know what's going to be going into each box. In each of these boxes then you're going to pull the angle or the side that is the congruent angle or side pair from the diagram. We know that this first one is going to be one of the angle relationships. I'll put angle two and angle four in the first box. The second box is going to be the side relationship and that's where AH is congruent to itself. And then the third box will be that second angle relationship, which on my diagram shows angle one is congruent to angle three. Anything going into these middle boxes again is always going to be an angle congruent to another angle or a side congruent to another side. From here, you can either work up top or down on the bottom. It doesn't matter. I'm going to start on the top first and start with the very top, which is my given statement. Math is a parallelogram. I know that that given statement is going to be at the very top and I'm going to put it here and that will help me to decide how I got these three congruent pieces. This given statement, math is a parallelogram, doesn't directly get me to say that angle two is congruent to angle four, AH is congruent to AH or angle one is congruent to angle three. So I need to use the definition or the postulate and theorems on my DPT sheet to get down to being able to say that these pieces are congruent. I know that because math is a parallelogram, I have two sets of parallel sides here, and so that's what I'm going to do first is put that if-then statement up there, and that comes right from your sheet. If I have a parallelogram, then the opposite sides are parallel. I know I have two sets of opposite sides, so I can go ahead and split that into each set of opposite sides. Remember this if-then statement in between the boxes is always going to be if this box, if I have a parallelogram, then this box, then the opposite sides are parallel. Make sure those if-then statements match up as you go along with your flow proof. Another important thing to point out here is that I have two sets of parallel lines and two sets of angles. You want to make sure that you're matching them up correctly. If I know that AT and MH are congruent, that leads me to angle two being congruent to angle four, because if these were the parallel lines, AT and MH, this would be the transversal, and then you'll see that angle A is included in between the parallel line and the transversal as is angle four. Angle one and three are not included, touch the parallel line to the transversal. The other parallel line, MH and TH, will give us the one in the three being congruent. Now that I have the two parallel lines or the two parallel sides, I can look on my DPT sheet and see that there is a theorem that says, if I have parallel lines, then I have congruent angles. And that's where you need to realize that these are alternate interior angles, and then the flow proof will come down. If parallel, then alternate interiors are angle. Alternate interior angles are congruent. Same thing on this side, if parallel lines, then the alternate interior angles are congruent. So that takes care of the two angle pieces up here. The last thing I have to do on top is to decide what led me to say that AH is congruent to AH, and that's where you need to remember that that's the reflex of property whenever you have something that is congruent to itself. I added in also this if then piece to bring those three congruent pieces together. If I have angle side angle, then my triangles are congruent. Remember, you don't necessarily have to put the A, the S, and the A there, but that will help you. So this piece is not going to be a mistake. If ASA, then the triangles are congruent. At this point, then, we're going to list the two congruent triangles, and it doesn't matter what order you write them out as, as long as whatever you pick for your first triangle, your corresponding pieces match up. I know that M is congruent to T, M is congruent to T, A, or not congruent corresponding, A is corresponding to H, and H is corresponding to A. And if it helps, you can just mark those off on your diagram up here to make sure that that is in the correct order. If we stop and remember what we were trying to prove, go back to your prove statement on the top, we're trying to prove not that the triangles are congruent, but that angle M is congruent to angle T, that those opposite angles are congruent. So our last step is to keep on going, and whenever we show that the triangles are congruent, that's where that CT comes in. Corresponding parts of congruent triangles are always congruent, and so whenever you're doing pieces of the triangles, you want that last step, and we are done with our proof. We are trying to prove that if a quadrilateral is a parallelogram, then opposite angles are congruent.