If the bisector of the vertical angle of a triangle bisects the base of the triangle. then the triangle is isosceles. GIVEN : A `A B C` in which `A D` is the bisector of `/_A` meeting `B C` in `D` such that `B D=D Cdot`

Explanation

Given:` A triangleABC `in which `AD` is the bisector of `angleA` meeting` BC` in `D` such that `BD = CD`
To Prove: `triangleACD` is an isosceles triangle.
Construction: We produce `AD` to `E` such that `AD = DE` and join `EC`.
Proof: In `triangleADB` and `triangleEDC` we have
`AD = DE` [By construction]
`angleADB = angleCDE `[Vertically opposite angles] `BD = DC` [Given]
By `SAS` criterion of congruence, we get
`triangleADB cong triangleEDC `
`AB = EC .......(i)` And,
`angleBAD = angleCED` [By cpctc]
But, `angleBAD = angleCAD `
`angleCAD = angleCED`
`AC = EC` [Sides opposite to equal angles are equal]
`AC = AB` [By eg. (i)]