Question

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

Answer 1

In ΔABC, BP and CP are bisectors of angles B and C respectively.
Hence ∠A + ∠B + ∠C = 180°
∠A + 2∠1 + 2∠2 = 180°
2(∠1 + ∠2) = 180° – ∠A
(∠1 + ∠2) = 90° – (∠A/2) --- (1)
In ΔPBC, ∠P + ∠1 + ∠2 = 180°
∠P + [90° – (∠A/2)] = 180° [From (1)]
∠P = 180° – [90° – (∠A/2)]
= [90° + (∠A/2)]
Hence angle P is always greater than 90°.
Thus PBC can never be a right angled triangle.