Diagonals of a Parallelogram are Equal & Intersect at Right Angles

If the diagonals of the parallelogram are the same and intersect at the right angles then that parallelogram is known as a square. Students who are looking at various sites to learn the theorems on parallelograms can go through this page and practice well for the exams. In this article, we will prove that the Diagonals of a Parallelogram are Equal & Intersect at Right Angles.

Diagonals of a Parallelogram are Equal & Intersect at Right Angles

Theorem:
The diagonals of a parallelogram are equal and intersect at right angles, then the parallelogram will be a square.
Given:
ABCD is a parallelogram in which AB ∥ DC, AD ∥ BC and diagonal AC are perpendicular to diagonal BD.
square_1
To prove that:
PQRS is a square, then AB = BC = CD = DA and an angle, say ∠DAB = 90°.
Proof:
In ∆ABC and ∆CDA,
∠BAC = ∠ACD (Since, AB ∥ DC and BC is a transversal)
∠BCA = ∠DAC (Since, BC ∥ AD and AC is a transversal)
AC = AC (Common Side).
Therefore, ∆ABC ≅ ∆CDA (By AAS criterion of congruency).
Therefore, AB = DC (CPCTC).
Similarly, ∆ABD ≅ ∆CDB (By AAS criterion of congruency).
Therefore, AD = BC (CPCTC).
∆OAB ≅ ∆OCD (By AAS criterion of congruency).
Therefore, OA = OC (CPCTC).
Similarly, ∆AOB ≅ ∆COB (By SAS criterion of congruency).
Therefore, AB = BC (CPCTC).
Therefore, AB = BC = CD = DA. (Proved)
∆DAB ≅ ∆CBA (By SSS criterion of congruency).
Therefore, ∠DAB = ∠CBA (CPCTC).
But ∠DAB + ∠CBA = 180° (Since, AD∥ BC).
Therefore, ∠DAB = ∠CBA = 180°/2 = 90°.
Hence proved.

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FAQs on Diagonals of a Parallelogram are Equal & Intersect at Right Angles

1. Do diagonals intersect at right angles?

In any rhombus, the diagonals bisect each other at right angles. That is, each diagonal cuts the other into two equal parts, and the angle where they cross is always 90 degrees.

2. Are the diagonals of a parallelogram always equal?

The diagonals of a parallelogram are not of equal length. They bisect with each other at the point of intersection with equal sides across the point of intersection.

3. Can a parallelogram have exactly 2 right angles?

A parallelogram is a quadrilateral with 2 pairs of opposite sides parallel. A rectangle is a special parallelogram that has 4 right angles.

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