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Math Help - Frustrating problem

  1. #1
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    Frustrating problem

    Sorry if this seems really simple... I haven't had geometry in school in over a year.

    The bisectors of angles B and C in triangle ABC intersect in point O. Find angle BOC if angle BAC = a (pretend that's an alpha).

    Bleh, I translated this from Russian, sorry if anything's unclear. Any help is appreciated though!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by pianistonstrike View Post
    Sorry if this seems really simple... I haven't had geometry in school in over a year.

    The bisectors of angles B and C in triangle ABC intersect in point O. Find angle BOC if angle BAC = a (pretend that's an alpha).

    Bleh, I translated this from Russian, sorry if anything's unclear. Any help is appreciated though!
    See the diagram below.

    I will not use three letters to describe an angle, i'll use one.

    Let O be angle BOC
    Let \alpha be angle BAC
    Let C be angle ACB
    Let B be angle ABC


    We want to find angle O (the red angle) in terms of \alpha

    Remember that all angles in a triangle add up to 180^{ \circ}, and that bisecting an angle means dividing it into two equal parts.

    So, for \triangle ABC we must have that:

    \alpha + B + C = 180

    \Rightarrow B + C = 180 - \alpha .................(1)

    Now for \triangle BOC we must have:

    O + \frac {1}{2}B + \frac {1}{2}C = 180

    \Rightarrow O = 180 - \frac {1}{2}B - \frac {1}{2}C

    \Rightarrow O = 180 - \frac {1}{2}(B + C)

    Now substitute the expression for B + C from equation (1), we get:

    O = 180 - \frac {1}{2} (180 - \alpha )

    \Rightarrow O = 180 - 90 + \frac {1}{2} \alpha

    \Rightarrow \boxed { O = 90 + \frac {1}{2} \alpha }
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  3. #3
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    Remember property of every concave quadrilateral and we have that (in Jhevon's sketch, where I denote \measuredangle~A=2\alpha)

    \measuredangle~\theta=2\alpha+\beta+\gamma=90^\cir  c+\frac{\measuredangle~A}2~\blacksquare
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  4. #4
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    Oh... duh, that makes perfect sense! I had gotten several of those steps but couldn't figure out how to tie them together.

    Thank you so much!
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