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.

 

[Hurkyl]

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]

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]

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]

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?