Two circles touch each other externally at point C. A common tangent AB touches the circles at points A and B respectively. Find the value of \(\angle ACB\).

Q.Two circles touch each other externally at point C. A common tangent AB touches the circles at points A and B respectively. Find the value of \(\angle ACB\).

AP = PC \[ \therefore\ \angle PAC = \angle PCA \] Similarly, \(\angle PBC = \angle PCB\) \[ \angle ACB = \angle PCB + \angle PCA = \angle PAC + \angle PBC \] In triangle ∆ABC: \[ \angle BAC + \angle ABC + \angle BCA = 180^\circ \Rightarrow \angle PAC + \angle PBC + \angle PAC + \angle PBC = 180^\circ \Rightarrow \angle PAC + \angle PBC = 90^\circ \] \[ \therefore\ \angle ACB = \angle PAC + \angle PBC = 90^\circ \]