The bisectors of the angles C and D of a parallelogram meet at the point P which lies on AB - Maths - Angles and Lines

Question

The bisectors of the angles C and D of a parallelogram meet at the
point P which lies on AB Prove that AB= 2AD

Varun Rawat · Accepted Answer

We have, ABCD as the given parallelogram
Since PA bisects ∠A and PB bisects ∠B
Now, ∠DAP = ∠PAB   and   ∠CBP = ∠PBASince, AB∥DC and PA is a transversal, then  ∠DPA = ∠PAB         Alternate interior angles⇒∠DPA = ∠DAP      as, ∠PAB =∠DAP ⇒DA = DP                   sides opposite to equal angles are equalSince, AB∥DC and PB is a transversal, then    ∠CPB = ∠ABP        Alternate interior angles⇒∠CPB = ∠CBP       as, ∠ABP =∠CBP ⇒CB = CP                    sides opposite to equal angles are equalNow, DC = DP + CP⇒AB = AD + CB          Opposite sides of ∥gm are equal⇒AB = AD + AD    ⇒AB = 2AD

The bisectors of the angles C and D of a parallelogram meet at the point P which lies on AB. Prove that AB= 2AD