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The bisectors of the angle of a parallelogram enclose a ____.

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Detailed Solution

The bisectors of the angle of a parallelogram enclose a rectangle.
Construct a parallelogram

A B C D

where

A B ‖ C D

and

A D ‖ B C

as shown in the figure below.

B E

bisects

∠ B

A E

bisects

∠ A

C G

bisects

∠ C

and

D G

bisects

∠ D

Consider, the adjacent angles

∠ A, ∠ B

of a parallelogram. The adjacent angles in a parallelogram are supplementary.

∠ A + ∠ B = 180 ° 12 ∠ A + 12 ∠ B = 90 °

∠ E A B + ∠ E B A = 90 °

......(1)
Consider

Δ E A B

, applying angle sum property of the triangle,

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

Since

∠ B E A

and

∠ F E H

are the opposite angles,
Therefore,

∠ F E H = 90 °

. [Opposite angles are equal]
Consider, similarly for

Δ B H C, Δ A F D, Δ C G D

∠ E F G = 90 ° ∠ G H E = 90 ° ∠ F G H = 90 °

Thus,

E F G H

is a rectangle as all opposite angles are

90 °

.
Hence, the required answer is rectangle.

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