Prove that the bisectors of any two consecutive angles of a rectangle are perpendicular to each other
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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°)
1/2BAD +1/2CDA = 1/2 180
⇒ ∠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.
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°)
1/2BAD +1/2CDA = 1/2 180
⇒ ∠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.
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