# Geometry: What Theorem Is This?

wubie
Hello,

I cannot remember what the theorem is in which the following happens:

Given two lines l and m which intersect each other, let H be the point of intersection.

Let A and B be points on the line l such that AHB are colinear. And let C and D be points on the line m such that CHD are colinear.

Now what is the theorem/lemma/corollary which states that when two such lines intersect in such a way that

angle AHC = angle BHD

and

angle AHD = angle BHC ?

I need to quote it for a proof that I am doing. I can't remember for my life. And I can't seem to find it in my notes/text. It's not a big problem, I would just like to quote it properly.

Any help is appreciated. Thankyou.

## Answers and Replies

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Hurkyl
Staff Emeritus
Gold Member
You also need A*H*B and C*H*D (H is between A and B, and H is between C and D). The theorem is called the vertical angles theorem.

Doc Al
Mentor
vertical angles

Don't know if it has a catchy name. How about the "vertical angles are equal" theorem? (The angles you mentioned are called vertical angles.)

edit: beaten again!

wubie
Thanks Hurkyl and Doc Al. By the way,

You also need A*H*B and C*H*D (H is between A and B, and H is between C and D).
Isn't that implied with the notation AHB and CHD and stating that they are colinear on their respective lines l and m?

Cheers.

Hurkyl
Staff Emeritus
Gold Member
I haven't seen such notation used before, but it certainly wouldn't surprise me that some would use it. As long as your teacher knows what it means.

HallsofIvy
Homework Helper
Actually, once you have said "Let A and B be points on the line l" and said that H is the point where the two lines intersect, it is not necessary to say (again) that they are "collinear". I don't believe that just saying "AHB are collinear" is a standard way of saying that H is between A and B.

wubie
I think I see what you mean.

Would simply stating

Given two lines l and m which intersect each other, let H be the point of intersection.

Let AHB be points on the line l and let CHD be points on the line m.

have been adequate then?