User:Valtururoshini/Sandbox

Theorem:

Angle bisectors of opposite angles of a parallelogram are parallel to each other.

Proof:

Let ABCD be a parallelogram. DE and BF are angle bisectors.

We have AB ll DC, AD ll BC. We know that angle ADE = angle EDF (DE is an angle bisector) ....(1)

angle EDF = angle DEA (Alternate interior angles)           ....(2)

Therefore, from (1) and (2), angle ADE = angle DEA = 1/2 angle ADC ....(a)

We also have angle FBC = angle FBE = 1/2 angle ABC But angle ABC = angle ADC.

Therefore angle FBE = 1/2 angle ADC ....(b) From (a) and (b), angle DEA = angle FBE (Corresponding angles when DE and BF are lines and AB is transversal)

Therefore, DE ll BF

Hence the theorem.