Question

Proove that the bisectors of two adjacent supplementary angles 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...