Prove that if the opposite sides of a quadrilateral are equal, the quadrilateral is a parallelogram.

Open in App

Solution

Given: Quadrilateral ABCD in which AB = DC and AD = BC$
To prove: ABCD is a parallelogram.
Construction: Draw diagonal AC.
Proof: Some statements and reasons for their validity are given below:

Statement Reason
AD=BC Given
AB=CD Given
AC=AC Common
$△ C B A ≅ △ A D C$ By SSS postulate
$∠ D C A = ∠ B A C$ Corresponding angles of congruent triangles
$D C ∥ A B$ Alternate angles are equal.
$∠ D A C = ∠ B C A$ Corresponding angles of congruent triangles
$A D ∥ B C$ Alternate angles are equal.
It follows that ABCD is a parallelogram.

Suggest Corrections

Similar questions

Prove that a quadrilateral with both pairs of opposite sides equal, is a parallelogram. [4 MARKS]

Prove that if one pair of opposite sides of a quadrilateral are equal and parallel, then it is a parallelogram.

A quadrilateral is a parallelogram, if pair of opposite sides are equal and __.

If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that is diagonals are equal.

Join BYJU'S Learning Program

Statement	Reason
AD=BC	Given
AB=CD	Given
AC=AC	Common
$△ C B A ≅ △ A D C$	By SSS postulate
$∠ D C A = ∠ B A C$	Corresponding angles of congruent triangles
$D C ∥ A B$	Alternate angles are equal.
$∠ D A C = ∠ B C A$	Corresponding angles of congruent triangles
$A D ∥ B C$	Alternate angles are equal.