Proof: Diagonals of isosceles trapezoid
About this video
Using the exclusive definition of a trapezoid (exactly one pair of parallel sides), the video proves that an isosceles trapezoid has congruent diagonals. Triangles CBA and DAB share side BA, have equal legs CB and DA, and have equal base angles between them, so they are congruent by SAS and CA = DB.
You'll be able to
- explain why the exclusive definition of a trapezoid is used for this proof.
- prove the diagonals of an isosceles trapezoid are congruent using SAS congruence.
Key words
- exclusive definition
- A trapezoid has only one pair of parallel sides.
- isosceles trapezoid
- A trapezoid whose legs are the same length.
- SAS congruence
- Two triangles are congruent if two sides and the angle between them match.
- diagonal
- A segment joining two opposite corners of a shape.
Transcript
- [Instructor] So I have an isosceles trapezoid here. And when we're talking about trapezoids, in this proof, and actually in some of the other videos that I've done alongside this, we're assuming the exclusive definition of trapezoids. Now, what do I mean by exclusive definition?
We're saying that a trapezoid or the definition we're gonna use has only one pair of parallel sides. There's an inclusive definition of trapezoids that could say, well, maybe you could have at least one pair of parallel sides, and then that would include things like parallelograms.
But we're not going to take that inclusive definition, the one that includes more things in it for the sake of this proof. And actually even the other video proofs that I've done along these lines, we're assuming the exclusive definition, that only one pair of parallel sides is how we're gonna define trapezoids here.
So assuming that as we have an isosceles trapezoid under that definition, what I wanna prove here is that the diagonals of that isosceles trapezoid are congruent. Essentially, I wanna prove that BD is equal to CA. So pause this video and have a go at that if you are so inspired.
All right, now let's work on this together. Well, in a previous video where we also assumed the inclusive definition of, or sorry, we also assumed the exclusive definition of trapezoids, we saw that the base angles of an isosceles trapezoid are going to be congruent.
So that angle is equal to that angle. Now notice triangle, let's look at the relationship between triangle CBA, triangle CBA and triangle DAB, DAB, CB corresponds to DA and they are congruent. They have an angle in between that are congruent, and then they both have this side right over here, which is of course equal to itself.
BA is on both of them. So by side angle side, these two triangles are congruent. Well, of these two triangles are congruent. We can see that the other corresponding side is also going to be, or corresponding sides, are also going to be congruent. So this side and this side are also going to be congruent.
So therefore we can say that CA, CA, is equal to DB, and we are done.
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