circle geometry

 

 

 

 

 

 

 

 

 

 

 

Given :2 circles with centrs O’ & O intersecting each other at A &B a tangent is drawn to circle with centre O’ at point A.

TPT: OA bisects angle CAB, i.e angle CAO = angle OAB

Construction : join OA,OB.

Proof:

1.     OA = OB……….          radii of the same circle

2.     angle OAB =OBA….. BASE ANGLES

3.     angle OAB +OBA+AOB =180^o    angle sum property

4.     2 OAB +AOB  = 180^o…….from (2)

5.     also angle CAO +OAB+BAD =180^o ……..CD is a straight line 

6.     equating  ( 4) & (5)  2 OAB +AOB =  CAO +OAB+BAD

7.     OAB +AOB = CAO +BAD

8.     but angle BAD = angle AOB………ANG IN ALTER SEGMENT

9.     angle OAB = CAO   from  (7)

      Hence proved