Given: ABC is a triangle . Let AB be produced to P and AC be produced to Q. Let the bisectors of ∠ PBC and ∠ QCB meet at O.
To prove: ∠ BOC = 90° -

Proof: In Δ ABC,
∠ A + ∠ B + ∠ C = 180° (Sum of the three angles of a triangle is 180° )

∠ A +

∠ B +

∠ C =

× 180°
∴

= 90° -

……………………….(1)

= 90° -
In Δ BOC,
∠ BOC + ∠ CBO + ∠ BCO = 180°
∠ BOC = 180° - (∠ CBO + ∠ BCO) ………………(2)
∠ PBC = 180° - ∠ ABC (Since ∠ ABC and ∠ CBP form a linear pair)
Dividing by 2,

∠ PBC =

(180° - ∠ ABC)
∴ ∠ CBO = 90°

…………………..(3)
Ð QCB = 180° - ∠ ACB (Since ∠ QCB and ∠ ACB form a linear pair)

∠ QCB =

(180° - Ð ACB)
∴ ∠ BCO = 90° -

………………………(4)
Substituting (3) and (4) in (2),
∠ BOC = 180° - (∠ CBO + ∠ BCO)
= 180° - (90° -

+ 90° -

)
= 180° - (180° -

-

)
=

+

…………………………….(5)
But from (1),

+

= 90° -
Substituting in (5),
∠ BOC = 90° -