#### AB is a diameter and AC is a chord of a circle with centre O such that $\angle$BAC = $30^{\circ}$. The tangent at C intersects extended AB at a point D. Prove that BC = BD.

Solution

Given AB is a diameter and AC is a chord of circle with center O.
$\angle BAC= 30^{\circ}$
To Prove : BC = BD
Construction : Join B and C
Proof :$\angle BCD= \angle CAB$[Angle is alternate segment]
$\angle BCD= 30^{\circ}\: \left [ Given \right ]$
$\angle BCD= 30^{\circ}\cdots \left ( i \right )$
$\angle ACB= 90^{\circ}$ [Angle in semi circle formed is 90°]
In $\bigtriangleup$ABC
$\angle CAB+\angle ABC+\angle BCA= 180^{\circ}$   [Sum of interior angles of a triangle is 180°]
$30^{\circ}+\angle ABC+90^{\circ}= 180^{\circ}$
$\angle ABC= 180^{\circ}-90^{\circ}-30^{\circ}$
$\angle ABC= 60^{\circ}$
Also , $\angle ABC+\angle CBD= 180^{\circ}$ [Linear pair]
$\angle CBD= 180^{\circ}-60^{\circ}$
$\angle CBD= 120^{\circ}$
$\angle ABC= 60^{\circ}$
In $\bigtriangleup CBD$
$\angle CBD+\angle BDC+\angle DCB= 180^{\circ}$
$\angle BDC= 30^{\circ}\cdots \left ( ii \right )$
From equation (i) and (ii)
$\angle BCD= \angle BDC$
$\Rightarrow BC= BD$[$\because$  Sides opposite to equal angles are equal]
Hence Proved