In a parallelogram show that the angle bisectors of two adjacent angles intersect at right angles

ABCD is a parallelogram. OA and OD are the bisectors of adjacent angles,∠A and ∠D.

ABCD is a parallelogram. 

∴ AB||DC    (Opposite sides of the parallelogram are parallel)

AB||DC and AD is the transversal,

∴ ∠BAD + ∠CDA = 180°  (Sum of interior angles on the same side of the transversal is 180°)

⇒ ∠1 + ∠2 = 90°    (AO and DO are angle bisectors ∠A and ∠D)  ...(1)

In ΔAOD,

∠1 + ∠AOD + ∠2 = 180°

⇒∠AOD + 90° = 180°  [from (1)]

⇒∠AOD  = 180° – 90° = 90°

∴ In a parallelogram, the bisectors of the adjacent angles intersect at right angle.

Cheers!