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#1

$Old$ November 7th, 2009, 07:33 AM

thereddevils $thereddevils is offline$

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In the diagram , X ,Y and Z are the points of contact of tangents BC , CA and AB respectively to a circle with centre O . If angle ACB is a right angle , prove that angle AOB =135 degree .

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$Old$ November 7th, 2009, 07:58 AM

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In the diagram , X ,Y and Z are the points of contact of tangents BC , CA and AB respectively to a circle with centre O . If angle ACB is a right angle , prove that angle AOB =135 degree .

Suppose $\angle BAO = x,\, \angle ABO = y$ .

Then

$\angle OAY = x$ (congruent $\triangle$ 's $OAZ, AOY$ )

$\angle OBX = y$ (congruent $\triangle$ 's $OBX, OBZ$ )

But

$2x + 2y = 90^o$ (angle sum of $\triangle ABC$ )

$\Rightarrow x + y = 45^o$

$\Rightarrow \angle AOB = 135^o$ (angle sum of $\triangle AOB$ )

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$Old$ November 7th, 2009, 07:44 PM

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Originally Posted by Grandad $View Post$

Hello thereddevilsSuppose $\angle BAO = x,\, \angle ABO = y$ .

Then

$\angle OAY = x$ (congruent $\triangle$ 's $OAZ, AOY$ )

$\angle OBX = y$ (congruent $\triangle$ 's $OBX, OBZ$ )

But

$2x + 2y = 90^o$ (angle sum of $\triangle ABC$ )

$\Rightarrow x + y = 45^o$

$\Rightarrow \angle AOB = 135^o$ (angle sum of $\triangle AOB$ )

Grandad

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