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Angle bisectors of a parallelogram form a ____.

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Detailed Solution

Angle bisectors of a parallelogram form a rectangle.
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 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 °

.
Hence,

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

Also, the sum of all the angles of a triangle is

180 °

.
Hence,

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

Similarly, in

Δ D P C

∠ P = 90 °

∴

All the angles formed by the bisectors of adjacent angles meet each other at

90 °

.
As all the four angles formed are right angles, so by the definition of rectangle, the angle bisectors of parallelogram form a rectangle.

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