Question

In the given figure, if ∠AOB = 80° and ∠ABC = 30°, then find ∠CAO. 

Answer 1

Consider the given circle with the centre ‘O’. Let the radius of this circle be ‘r’. ‘AB’ forms a chord and it subtends an angle of 80° with its centre, that is`angleAOB ` = 80°.

The angle subtended by an arc at the centre of the circle is double the angle subtended by the arc in the remaining part of the circle.

So, here we have

`angleACB = ( angleAOB )/2`

            `=(80°)/2`

  `angleACB = 40°`

In any triangle the sum of the interior angles need to be equal to 180°.

Consider the triangle  ΔAOB

`angleAOB + angleOAB + angleOBA = 180°`

Since,  `OA = OB = r , we have `angle OAB = angleOBA `. So the above equation now changes to

`angleAOB + angleOAB + angleOAB ` = 180°

                                 `2 angle OAB = 180° - angleAOB `

                                               = 180° - 80°

                                 `2angleOAB` = 100°

                                    `angleOAB ` = 50 °

Considering the triangle ΔABC now,

`angleACB + angleOAB + angle OAC + angleABC ` = 180°

                                                    `angle OAC = 180° - angleACB - angleOAB - angleABC` 

                                                                  = 180°- 40°- 50° - 30° 

                                                    `angleOAC`  =  60°

Hence, the measure of  `angleCAO `   is  60° .