Question

prove that the bisectors of the angles of a parallelogram enclose a rectangle

Answer 1

Given: ABCD is a parallelogram. AE bisects ∠BAD. BF bisects ∠ABC. CG bisects ∠BCD and DH bisects ∠ADC

To prove: LKJI is a rectangle

∠BAD + ∠ABC = 180° because adjacent angles of a parallelogram are supplementary

[Since sum of adjacent angles of a parallelogram are supplementary]

ΔABJ is a right triangle since its acute interior angles are complementary

Similar in ΔCDL we get ∠DLC = 90° and in ΔADI we get ∠AID = 90°

Then ∠JIL = 90° as ∠AID and ∠JIL are vertical opposite angles

Since three angles of quadrilateral LKJI are right angles, hence 4th angle is also a right angle.

Thus LKJI is a rectangle.