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Geometry: What Theorem Is This?

  1. Nov 18, 2003 #1
    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.
     
  2. jcsd
  3. Nov 18, 2003 #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.
     
  4. Nov 18, 2003 #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!
     
  5. Nov 18, 2003 #4
    Thanks Hurkyl and Doc Al. By the way,

    Isn't that implied with the notation AHB and CHD and stating that they are colinear on their respective lines l and m?

    Cheers.
     
  6. Nov 18, 2003 #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:
     
  7. Nov 18, 2003 #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.
     
  8. Nov 19, 2003 #7
    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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