Screen capture video: Prove that a quadrilateral is a parallelogram
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This mathematical screen-capture video, created by high-school math teacher Tina Zack, demonstrates the process of proving that a quadrilateral is a parallelogram. The video was recorded using a Mobi device.
- Christina Zack (00:01)
- Okay, this is number four from the proof worksheet, proving a quad is a parallelogram. All right, we’re trying to prove that ABZX is a parallelogram. Now, I’ve marked all of the given information, but if I can prove these triangles congruent first, which I will do that, and use their parts to get parts of this quadrilateral that I need, that would be one way of going about doing this proof.
- So I only put in the part of the given that I’m actually going to use to prove these triangles are congruent. And of course, I can mark that angle one and two are congruent because these are vertical angles, so I think that’s what I’ll go ahead and put down next, that angle one is congruent to angle two. That’s because vertical angles are congruent.
- Alright, now I can say that these triangles are congruent. I’m going to name the triangle “Triangle BYZ.” If I name it like that, the other triangle is going to be CYX. And that’s because of the side angle side postulate. Okay, I’m going to use the parts of those triangles.
- The parts that I’m going to use is going to be ZB and CX. And that’s because corresponding sides of congruent triangles are congruent. Now, in the given, I did not write this down as step one. I could have. I know that AX is congruent to CX, so that was again, given, and it could be step one. I think it’s easier to realize now that I can use the transitive property and say that ZB is congruent to AX, and that’s just the transitive property of congruence.
- All right. Now that will give me these congruent here. So I’m just going to mark them again on my diagram. So I can see I’ve got one pair of opposite sides of congruent. Well if I can get those pair of opposite sides parallel, then that is one of the combo methods for proving that a quad is a parallelogram.
- So if I look in my triangles, I see that three and four. Now three and four are going to be corresponding angles because they would lie on top of one another. They would match up. So I can now say that — and I could have put this in step four, but I’m separating it — angle three is congruent to angle four. And that’s because corresponding angles of congruent triangles are congruent.
- Well that actually makes Z, and it gives me parallel lines, so that means that ZB is parallel to CA. And so, I can say that if two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
- All right. Even though XA is the side I want to prove parallel to ZB, since XA is a part of CA, I can say now that indeed, this has to be a parallelogram, because when I mark my figure, I’m marking these right here. And so XA is a part of CA, so to me this is the combo method. So that proves that this quad, ABZX is a parallelogram.
- So I’m going to write ABZX is a parallelogram, and the reason is going to be the combo method which states: If a quad has one pair of opposite sides both parallel and congruent, then it is a parallelogram. So that is one way to do that proof. It is not the only way to do this. If you just put in five with the given, then obviously it would only be eight steps. Hopefully that helped.