Let us prove that

Isosceles trapezium is a trapezium in which th non parallel sides are equal

Consider ABCD as the isosceles trapezium with AD = BC

We have to prove that ∠D = ∠C

Drop perpendiculars from point A and B on CD at points E and F respectively as shown

Consider ΔAED and ΔBFC

AD = BC … given … (i)

AE = BF … perpendiculars between two parallel lines … (ii)

Using Pythagoras theorem

DE² = and FC² =

Using (i) and (ii)

⇒ DE² =

⇒ DE² = FC²

⇒ DE = FC … (iii)

Using (i), (ii) and (iii)

Therefore, ΔAED ≅ ΔBFC

⇒ ∠ADE = ∠BCF … corresponding angles of congruent triangles

⇒ ∠D = ∠C

Hence proved