CK12 - Geometry

Conversestatementsare importantin geometry. It is crucialto knowwhichtheoremshavetrue converses.
In the caseof parallelograms,almostall of the theoremsyou havestudiedthis far havetrue converses.This
lessonexploreswhichcharacteristicsof quadrilateralsensurethat they are parallelograms.

Provinga Quadrilateralis a ParallelogramGivenCongruentSides

In the last lesson,you learnedthat a parallelogramhas congruentoppositesides.We provedthis earlier
and then lookedat one exampleof this usingthe distanceformulaon a coordinategrid to verifythat opposite
sidesof a parallelogramhad identicallengths.

Here,we will showon the coordinategrid that the converseof this statementis also true: If a quadrilateral
has two pairsof oppositesidesthat are congruent,then it is a parallelogram.

Example 1

Showthat the figureon the grid belowis a parallelogram.

We can see that the lengthsof oppositesidesin this quadrilateralare congruent.For example,to find the

lengthof we can find the differencein the -coordinates(6-1 = 5) because is horizontal(it’s

generally very easy to find the length of horizontaland vertical segments). and

. So, we haveestablishedthat oppositesidesof this quadrilateralare congruent.

But is it a parallelogram? Yes. Oneway to argue that CDEFis a parallelogram is to notethat

. We can think of as a transversalthat crosses and. Now,
interiorangleson the sameside of the transversalare supplementary, so we can applythe postulateif interior
angleson the sameside of the transversalare supplementarythen the lines crossedby the transversalare
parallel.

Note:This exampledoesnotprovethat if oppositesidesof a quadrilateralare congruentthen the quadrilateral
is a parallelogram.To do that you needto use any quadrilateralwith congruentoppositesides,and then
you use congruenttrianglesto help you. We will let you do that as an exercise,but here’s the basicpicture.

Whattrianglecongruencepostulatecan you use to show