The bisectors of base angles of a triangle cannot enclose a right angle in any case.

Explanation

In `△ABC`, `BO` and `CO` are bisectors of `angleB` and `angleC` respectively.
Now, in `△ABC`,
`∠A+∠B+∠C=180°`
`∠B+∠C=180°−∠A` .....(1)
Now, in `△BOC`,
`∠BOC+1/2∠B+1/2∠C=180°`
`∠BOC+1/2(∠B+∠C)=180°`
`∠BOC+1/2(180°−∠A)=180°`
`⇒∠BOC=180°−90°+1/2∠A`
`⇒∠BOC=90°+1/2∠A`
`⇒∠BOC>90°`
Thus it is proved that the bisector of base angles of a triangle cannot enclose a right angle.