Lesson Proof of Opposite sides of a parallelogram are equal

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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.

<b>Theorem</b>: If ABCD is a <A HREF=http://www.algebra.com/algebra/homework/Rectangles/Different-between-parallelogram-rectangle-square-rhombus-and-trapezoid.lesson>parallelogram</A> then prove that its opposite sides are equal.

{{{drawing( 160, 160, -10, 10, -10, 10, line( -6, -6,4,-6) , line(-6,-6,8,2),line( -6, -6,-2,2), line( -2, 2,8,2) , line( 8, 2,4,-6) ,locate( -6.5,-6.5,A),locate( 4.5,-6.5,B),locate( 8.5,3.5,C),locate(-2.5,4,D))}}}

<b>Proof</b>:

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  <A HREF=http://www.algebra.com/algebra/homework/Triangles/Triangles-and-its-basic-properties.lesson>Triangles </A> ABC and CDA as shown in figure.

In triangles,

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

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

3. {{{AC = CA}}}                             ......(The common side to two triangles)

From conditions 1,2 and 3,

Triangles ABC and CDA are congruent (By <A HREF=http://www.algebra.com/algebra/homework/Triangles/Angle.wikipedia>Angle </A>-Side-Angle congruency property)

Hence as triangles are <A HREF=http://www.algebra.com/algebra/homework/Triangles/Geometry-Proving-Triangles-Congruent.lesson>congruent triangles </A>, the corresponding sides are equal,

so AB = CD and BC = DA.

QED

To learn more about Similar and congruent triangles you can refer to <A HREF=http://www.algebra.com/algebra/homework/Triangles/Triangles-congruence-and-similarity.lesson>wikipedia </A>article.

To learn more about parallelogram you can refer to <A HREF=http://www.algebra.com/algebra/homework/Parallelograms/Parallelogram.wikipedia>wikipedia</A>