# Lesson Properties of diagonals of a parallelogram

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## Properties of diagonals of a parallelogram

Theorem 1
If in a quadrilateral both pairs of the opposite sides are parallel, then the two diagonals bisect each other. Prove.
 Proof You are given the quadrilateral ABCD (Figure 1) with two                   pairs of parallel sides: AB is parallel to DC and AD is parallel to BC. You need to prove that the diagonals AC and BD bisect each other, in other words, that the segments AP and PC, BP and PD are congruent: AP = PC, BP = PD, where P is the intersection point of the diagonals AC and BD. Quadrilaterals that have both pairs of the opposite sides parallel are called parallelograms. They were introduced and studied in the lesson Properties_of_the_sides_of_parallelograms under the topic Triangles in the section Geometry in this site. Figure 1. To the Theorem 1
It was proved in that lesson that the opposite sides of such quadrilaterals are congruent. In particular, for our quadrilateral ABCD the opposite sides AB and DC are congruent.

Furthermore, the angles BAC and ACD are congruent as they are the alternate interior angles at the parallel lines AB and DC and the transverse AC (see the lesson Parallel lines
in this site). The angles ABD and BDC are congruent too as they are the alternate interior angles at the parallel lines AB and DC and the transverse DB (see the same lesson
Parallel lines in this site). Hence, the triangles ABP and CDP are congruent in accordance to the postulate 2 (ASA) of the lesson Congruence tests for triangles, which is under the topic Triangles in the section Geometry in this site.

This means that the segments AP and PC, BP and PD are congruent: AP = PC, BP = PD as the corresponding sides of the congruent triangles ABP and CDP.

This is what to be demonstrated. The proof is completed.

The opposite statement to the Theorem 1 is valid too.

Theorem 2
If in a quadrilateral the two diagonals bisect each other, then both pairs of the opposite sides are parallel. Prove.
 Proof You are given the quadrilateral ABCD (Figure 2), in which                    the two diagonals AC and BD bisect each other. In other words, the segments AP and PC, BP and PD are congruent: AP = PC, and BP = PD, where P is the intersection point of the diagonals AC and BD. You need to prove that the opposite sides are parallel: AB is parallel to DC and AD is parallel to BC. The proof is actually very simple (Figure 3). Consider the triangles ABP and CDP. By the condition, they have congruent sides AP and PC, BP and PD. In addition, the included angles APB and CPD are congruent as vertical angles formed by the straight lines AC and BD. Figure 2. To the Theorem 2 Figure 3. To the proof of the Theorem 2

Therefore, the triangles ABP and CDP are congruent in accordance to the postulate 1 (SAS) of the lesson Congruence tests for triangles, which is under the topic Triangles
of the section Geometry in this site. This implies that the angles CAB and ACD are congruent as the corresponding angles of the congruent triangles ABP and CDP.
But these angles are the alternate interior angles formed by the straight lines AB and DC and the transverse AC (see the lesson Parallel lines under the topic of the section Geometry in this site). Hence, the straight lines AB and DC are parallel in accordance to the Converse of parallel transversal theorem of the same lesson Parallel lines.

Similarly, the triangles ADP and CBP (Figure 3) have congruent sides AP and PC, BP and PD by the condition. In addition, the included angles APD and CPB are congruent as
vertical angles formed by the straight lines AC and BD. Therefore, the triangles ADP and CBP are congruent in accordance to the postulate 1 (SAS) of the lesson
Congruence tests for triangles in this site. This implies that the angles APD and CPB are congruent as the corresponding angles of the congruent triangles ADP and CBP.
But these angles are the alternate interior angles formed by the straight lines AD and BC and the transverse BD (see the lesson Parallel lines in this site). Hence, the straight
lines AD and BC are parallel in accordance to the Converse of parallel transversal theorem of the same lesson Parallel lines.

Thus, we proved that AB is parallel to DC and AD is parallel to BC, exactly as the Theorem 2 requires.

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