Frustrating problem

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June 29th 2007, 07:09 PM
pianistonstrike

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!
June 29th 2007, 07:39 PM
Jhevon

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Quote:

Originally Posted by pianistonstrike

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:

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

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

$\Rightarrow \boxed { O = 90 + \frac {1}{2} \alpha }$
June 29th 2007, 07:49 PM
Krizalid

Remember property of every concave quadrilateral and we have that (in Jhevon's sketch, where I denote $\measuredangle~A=2\alpha$ )

$\measuredangle~\theta=2\alpha+\beta+\gamma=90^\cir c+\frac{\measuredangle~A}2~\blacksquare$
June 29th 2007, 07:54 PM
pianistonstrike

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!