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Math Help - Angles in a circle

  1. #1
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    Angles in a circle

    http://s14.photobucket.com/albums/a3...rrent=scan.jpg


    How would I prove that angle 2 is twice the size of angle 1?


    thankyou ^^
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by Vashti
    http://s14.photobucket.com/albums/a3...rrent=scan.jpg


    How would I prove that angle 2 is twice the size of angle 1?


    thankyou ^^
    firstly we will label the diagram
    let the vertex with angle 1 be A
    the two other unlabelled points be B and C
    join OA and extend it
    if you could follow the instructions upto here, then post a new diagram and I will give you the rest of the proof
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  3. #3
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    It is challenging to write out the proof without a diagram by the basic idea is this: If one of the lines that form that inscribed angle passes though the diameter then it clearly is true, why? Because the central angle is the exterior angle of the triangle which you have. But by the theorem: an exterior angle is the the sum of the non-abjacent angles of the triangle. Since the two non-adjacent angle are the same, because you have an isoseles. You can conlude it is twice as large as the inscribed angle.

    In the case why the center does not line on one of those line, unlike in first paragraph. You need to break the problem as a sum of two arc (if center is included between the lines) and difference of big arc and small arc (if center is not included between the arcs).

    Hope this helps.
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    Esther from the Bible?
    Last edited by ThePerfectHacker; May 31st 2006 at 06:21 PM.
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  4. #4
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    http://s14.photobucket.com/albums/a3...t=scan0001.jpg

    like this?
    (thankyou very much ^^)
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  5. #5
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    Quote Originally Posted by Vashti
    Hello,

    I've attached a diagram to explain how I labeled vertices, angles, etc.

    The central line CO generates 2 isosceles triangles: AOC and BOC. I've marked the angles at the base of each triangle.

    The angle \angle(KOB) = \angle{(c_1)}+\angle{(b_1)} and \angle(KOA) = \angle{(c_2)}+\angle{(a_2)}. Thus

    \angle(AOB) = \angle{(c_1)}+\angle{(b_1)}+\angle{(c_2)}+\angle{(  a_2)}=2 \cdot \angle(ACB)

    quod era demonstrandum

    Greetings

    EB
    Attached Thumbnails Attached Thumbnails Angles in a circle-puz1.gif  
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