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In a parallelogram, the angle bisectors of two adjacent angles intersect at ____.

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Detailed Solution

In a parallelogram, the angle bisectors of two adjacent angles intersect at 90°.
Suppose the parallelogram be ABCD whose bisectors of adjacent angles A, B intersect at O and of C, D bisect at P.
In

Δ A O B

, AO is the bisector of A, and hence,

∠ O A B = ∠ A 2

.
BO is the bisector of B hence,

∠ O B A = ∠ B 2

The sum of the adjacent angles of a parallelogram is

180 °

∠ A + ∠ B = 180 ° ∠ A + ∠ B 2 = 180 ° 2 ∠ A + ∠ B 2 = 90 °

As the sum of all the angles of a triangle is

180 °

. Hence, in

Δ A O B

∠ O A B + ∠ O B A + ∠ B O A = 180 ° ∠ B O A + 90 ° = 180 ° ∠ B O A = 90 °

Hence, the angle bisectors intersect each other at

90 °

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