Question

prove that the bisector of two adjacent supplementary angle include a right angle​

Answer 1

Solution:-

Given, ∠ DAB + EBA = 180°. CA and CB are bisectors of ∠ DAB ∠ EBA respectively.

∴ ∠ DAC + ∠ CAB = 1/2 (∠ DAB).....(1)

⇒ ∠ EBC + ∠ CBA = 1/2 (∠ EBA)....(2)

⇒ ∠ DAB + ∠ EBA = 180°

⇒ 2 (∠ CAB) + 2 (∠ CBA) = 180°  [using (1) and (2)]

⇒ ∠ CAB + ∠ CBA = 90°

In Δ ABC,

∠ CAB + ∠ CBA + ∠ ABC = 180° (Angle Sum property)

⇒ 90° + ∠ ABC = 180°

⇒ ∠ ABC = 180° - 90°

⇒ ∠ ABC = 90°

So, the bisectors of the two adjacent supplementary angles include a right angle.

Hence proved.