Question

Prove that the diagonals of an isosceles trapezium are equal.

Answer 1

Let us consider an isosceles trapezium PQRS such that PQ || SR and PS = QR.

PR and QS are its diagonals.

Construction: Draw two perpendiculars, PA and QB on the side SR.

In ΔPSA and ΔQRB, we have:

PS = QR (Given)

∠ PAS = ∠QBR = 90° (By construction)

PA = QB (Perpendicular distance between two parallel lines)

∴ ΔPSA ≅ ΔQRB (By RHS congruence criterion)

⇒ ∠PSA = ∠QRB (By c.p.c.t)

⇒∠PSR = ∠QRS

In ΔPSR and ΔQRS, we have:

PS = QR (Given)

∠PSR = ∠QRS (Proved above)

SR = RS (Common)

∴ ΔPSR ≅ ΔQRS (By SAS congruence criterion)

⇒ PR = QS (By c.p.c.t)

Hence, the diagonals of an isosceles trapezium are equal.