ABCD is trapezium in which AB || CD. If AD = BC, show that ∠A = ∠B and ∠C = ∠D.

Given:- ABCD is a isosceles trapezium

Trapezium with one pair of sides is parallel

And other pair of sides is equal[AD=BC].

Formula used:- SSS congruency property

If all 3 sides of triangle are equal to all 3 sides of

other triangle

Then; Both triangles are congruent

Solution:- In trapezium ABCD

If AB||CD

And; AD=BC

∴ ABCD is a isosceles trapezium

⇒ If ABCD is a isosceles trapezium

Then;

Diagonals of ABCD must be equal

∴ AC=BD

In Δ ABD and Δ ABC

AC=BD [∵ ABCD is isosceles trapezium]

AD=BC [Given]

AB=AB [Common line in both triangle]

∴ Both triangles are congruent by SSS property

∆ABD ≅ ∆ABC

⇒ If both triangles are congruent

Then there were also be equal.

∴ ∠ A=∠ B

As ABCD is trapezium and AB||CD

∠ A+∠ D=180° and ∠ C+∠ B=180°

∠ A =180° - ∠ D and ∠ B=180° - ∠ C

If ∠ A=∠ B;

180° - ∠ D=180° - ∠ C

180° +∠ C - 180° = ∠ D

∴ ∠ C=∠ D;

Conclusion:- If ABCD is trapezium in which AD = BC, then ∠A = ∠B and ∠C = ∠D.