Question

prove that the bisector of 2 adjacent supplementary angle include a right angle

Answer 1

so the bisector of two adjacent supplementary angles include a right angle
Given an. AOC and COB
two supplement angles
Rays OE and OD bisects ang BOC and AOC
TO PROVE== ang.DOE 90
ang. AOC+COB = 180
1/2AOC +1/2 COB =1/2 ×180 {OD BISECT AOC}
1/2 AOC +1/2 COB =90 {OE BISECT COB }

ang. DOC + ang. COE = 90
ang DOE = 90
PROVED

Answer 2

Solution:-
Given; angleDAB+EBA=180°. CA and CB are bisectors of angleDAB, angleEBA respectively.
So, angleDAC+angleCAB=1/2(angleDAB) ---(i)
=> angleEBC+angleCBA=1/2(angleEBA) ----(ii)
=>angleDAB+angleangleEBA=180°
=>2(angleCAB)+2(angleCBA)=180° [Using (i) and (ii)
=>angleCAB+angleCBA=90°
In triangleABC,
angleCAB+angleCBA+angleABC=180° [angle sum property)
=> 90°+angleABC=180°
=>angleABC=180-90
=>angleABC=90
So, the bisectors of the two adjacent supplentary angles include a right angle.
Hence proved
Hope it helps