1. ## Geometry (1)

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 .

2. Hello thereddevils
Originally Posted by thereddevils
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 $\displaystyle \angle BAO = x,\, \angle ABO = y$.

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

$\displaystyle \angle OBX = y$ (congruent $\displaystyle \triangle$'s $\displaystyle OBX, OBZ$)
But
$\displaystyle 2x + 2y = 90^o$ (angle sum of $\displaystyle \triangle ABC$)

$\displaystyle \Rightarrow x + y = 45^o$

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

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

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

$\displaystyle \angle OBX = y$ (congruent $\displaystyle \triangle$'s $\displaystyle OBX, OBZ$)
But
$\displaystyle 2x + 2y = 90^o$ (angle sum of $\displaystyle \triangle ABC$)

$\displaystyle \Rightarrow x + y = 45^o$

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