In an isosceles-trapezium, show that the opposite angles are supplementary.

ABCD is an isosceles trapezium in which AD = BC.

To prove:

∠ A + ∠ C = 180°

and, ∠ B + ∠ D = 180°

Proof:

AB is parallel to CD. So, sum of adjacent angles is 180°.

⇒ ∠ A + ∠ D = 180°

It is already given that ABCD is an isosceles trapezium which means AD = BC.

⇒ ∠ A = ∠ B

So,

⇒ ∠ B + ∠ D = 180°

In a trapezium, sum of all angles is always equal to 360°.

⇒ ∠ A + ∠ B + ∠ C + ∠ D = 360°

⇒ ∠ A + ∠ C + (∠ B + ∠ D) = 360°

⇒ ∠ A + ∠ C + 180° = 360°

⇒ ∠ A + ∠ C = 360° - 180°

⇒ ∠ A + ∠ C = 180°

Hence, the opposite angles are supplementary.