Lesson Properties of the sides of a parallelogram

Properties of the sides of a parallelogram

Let me remind you that a parallelogram is a quadrilateral which has both pairs of the opposite sides parallel.


Theorem 1
In a parallelogram, the opposite sides are of equal length in pairs.

Proof Let ABCD be a parallelogram (Figure 1) with the sides AB, BC, DC              and AD. We need to prove that the opposite sides AB and DC are of equal length, as well as the sides BC and AD. Let us draw the diagonal BD in the parallelogram ABCD and consider the triangles ABD and DCB (Figure 2). The diagonal BD is the common side of these triangles. The angles ABD and BDC are congruent as the alternate interior angles at the parallel lines AB and DC and the transverse BD. The angles

Figure 1. To the Theorem 1      Figure 2. To the proof of the Theorem 1

ADB and DBC are congruent as the alternate interior angles at the parallel lines AD and BC and the transverse BD.

Hence, the triangles ABD and DCB are congruent in accordance to the postulate P2 (ASA) of the lesson Congruence tests for triangles, which is under the topic Triangles
of the section Geometry in this site.
This implies that the sides AB and DC are of equal length as the corresponding sides the triangles ABD and DCB.
Similarly, the sides BC and AD are of equal length by the same reason.

The proof is completed.

The converse statement is valid too.

Theorem 2
If in a convex quadrilateral the opposite sides are of equal length by pairs, then the quadrilateral is a parallelogram.

Proof The Figure 3 shows a convex quadrilateral ABCD with the sides AB,              BC, DC and AD. The opposite sides AB and DC are of equal length: AB = DC. The opposite sides AD and BC are of equal length too: AD = BC. We need to prove that the quadrilateral ABCD is a parallelogram. Draw the diagonal BD in the parallelogram ABCD and consider the triangles ABD and DCB (Figure 4). These triangles have the congruent sides AB and DC by the condition. The sides AD and BC are congruent by the condition, too. The side BD is the common side.

Figure 3. To the Theorem 2      Figure 4. To the proof of the Theorem 2

Hence, the triangles ABD and DCB are congruent in accordance to the postulate P3 (SSS) of the lesson Congruence tests for triangles, which is under the topic Triangles
in the section Geometry in this site.

This means that the angles ABD and BDC are congruent as the corresponding angles of the congruent triangles ABD and DCB.
Hence, the straight lines AD and BC are parallel as they have congruent alternate interior angles ABD and BDC.

This also means that the angles ABD and DBC are congruent as the corresponding angles of the congruent triangles ABD and DCB.
Hence, the straight lines AB and BC are parallel as they have congruent alternate interior angles ADB and DBC.

The proof is completed.