proof of parallelogram theorems


Theorem 1.

The opposite sides of a parallelogramMathworldPlanetmath are congruent.

Proof.

This was proved in the parent (http://planetmath.org/ParallelogramTheorems) article. ∎

Theorem 2.

If both pairs of opposite sides of a quadrilateralMathworldPlanetmath are congruent, the quadrilateral is a parallelogram.

Proof.

Suppose ABCD is the given parallelogram, and draw AC¯.

Then ABCADC by SSS, since by assumptionPlanetmathPlanetmath AB=CD and AD=BC, and the two trianglesMathworldPlanetmath share a third side.

By CPCTC, it follows that BACDCA and that BCADAC. But the theorems about corresponding angles in transversal cutting then imply that AB¯ and CD¯ are parallelMathworldPlanetmathPlanetmath, and that AD¯ and BC¯ are parallel. Thus ABCD is a parallelogram. ∎

Theorem 3.

If one pair of opposite sides of a quadrilateral are both parallel and congruent, the quadrilateral is a parallelogram.

Proof.

Again let ABCD be the given parallelogram. Assume AB=CD and that AB¯ and CD¯ are parallel, and draw AC¯.

Since AB¯ and CD¯ are parallel, it follows that the alternate interior angles are equal: BACDCA. Then by SAS, ABCADC since they share a side.

Again by CPCTC we have that BC=AD, so both pairs of sides of the quadrilateral are congruent, so by Theorem 2, the quadrilateral is a parallelogram. ∎

Theorem 4.

The diagonals of a parallelogram bisect each other.

Proof.

Let ABCD be the given parallelogram, and draw the diagonals AC¯ and BD¯, intersecting at E.

Since ABCD is a parallelogram, we have that AB=CD. In addition, AB¯ and CD¯ are parallel, so the alternate interior angles are equal: ABDBDC and BACACD. Then by ASA, ABECDE.

By CPCTC we see that AE=CE and BE=DE, proving the theorem. ∎

Theorem 5.

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Proof.

Let ABCD be the given quadrilateral, and let its diagonals intersect in E.

Then by assumption, AE=EC and DE=EB. But also vertical angles are equal, so AEDAEB and CEDAEB. Thus, by SAS we have that AEDCEB and CEDAEB.

By CPCTC it follows that AB=CD and that AD=BC. By Theorem 1, ABCD is a parallelogram.

Title proof of parallelogram theorems
Canonical name ProofOfParallelogramTheorems
Date of creation 2013-03-22 17:15:48
Last modified on 2013-03-22 17:15:48
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 9
Author rm50 (10146)
Entry type Proof
Classification msc 51M04
Classification msc 51-01
Related topic Parallelogram