Lesson Proof of Opposite sides of a parallelogram are equal

In this lesson we will prove the basic property of a parallelogram that the opposite sides in a parallelogram are equal. The converse is also true that if opposite sides of a quadrangle are equal then its a parallelogram.

Theorem: If ABCD is a parallelogram then prove that its opposite sides are equal.





Proof:
By Parallelogram definition, line AB is parallel to line CD and line BC is parallel to line DA.

We are required to prove that AB = CD and BC = DA.

Let AC be the diagonal of parallelogram.

Consider two Triangles ABC and CDA as shown in figure.

In triangles,

1. .....(The line AC is a transversal of parallel lines AB and CD,hence Angle CAB and ACD are alternate angles)

2. .....(The line AC is a transversal of parallel lines BC and DA,hence Angle ACB and Angle CAD are alternate angles)

3. ......(The common side to two triangles)

From conditions 1,2 and 3,

Triangles ABC and CDA are congruent (By Angle -Side-Angle congruency property)

Hence as triangles are congruent triangles , the corresponding sides are equal,

so AB = CD and BC = DA.

QED

To learn more about Similar and congruent triangles you can refer to wikipedia article.

To learn more about parallelogram you can refer to wikipedia