Diagonals Converse: The Proof that a Quadrilateral is a Parallelogram when its Diagonals Bisect Each Other

Parallelogram Diagonals Converse

The diagonals of a parallelogram have a special property known as the diagonals converse

The diagonals of a parallelogram have a special property known as the diagonals converse. The diagonals converse states that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

To understand the diagonals converse, let’s consider a quadrilateral ABCD. Let the diagonals intersect at point E, such that AE = EC and BE = ED. We need to prove that ABCD is a parallelogram.

To prove that ABCD is a parallelogram, we must show that opposite sides are parallel.

Proof:

Since AE = EC and BE = ED, we can consider the triangles AEB and CED.

In triangle AEB, we have AE = BE, and in triangle CED, we have CE = DE. Also, since diagonals bisect each other, we have AE = EC and BE = ED.

Using the Side-Angle-Side (SAS) congruence criterion, we can conclude that triangle AEB and triangle CED are congruent.

Therefore, angle AEB is congruent to angle CED (by CPCTC, corresponding parts of congruent triangles are congruent).

Now, let’s examine angles AED and CEB.

Since angle AED and angle CEB are opposite angles formed by intersecting lines, they are congruent.

Now, we have angle AEB congruent to angle CED and angle AED congruent to angle CEB.

If we consider the transversal line AB, we can conclude that:

angle AEB + angle CEB = 180 degrees (since they are supplementary angles), and
angle AED + angle CED = 180 degrees (since they are supplementary angles).

Now, using the property of a straight line that the sum of angles forming a straight line is 180 degrees, we can equate:

angle AEB + angle CEB = angle AED + angle CED

Since angle AEB and angle CED are congruent, and angle AED and angle CEB are congruent, we rewrite the equation as:

angle AEB + angle CED = angle CEB + angle CED

Simplifying, we have:

angle AEB = angle CEB

This shows that the opposite angles of ABCD are congruent, which is a property of a parallelogram.

Therefore, by proving that opposite angles are congruent, we have demonstrated that ABCD is a parallelogram.

Hence, we have proved the diagonals converse: if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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