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Geometry: What Theorem Is This?

  • Thread starter wubie
  • Start date
  • #1
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

  • #2
Hurkyl
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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.
 
  • #3
Doc Al
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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!
 
  • #4
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.
 
  • #5
Hurkyl
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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. :smile:
 
  • #6
HallsofIvy
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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.
 
  • #7
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?
 

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