A parallelogram with a right angle is known as a rectangle. We know that the opposite sides of a parallelogram and rectangle are equal. The diagonals of the parallelogram are of equal lengths. Thus a parallelogram whose diagonals are of equal length is a rectangle. Opposite sides of a parallelogram and rectangle are always congruent. Hence we can say that a parallelogram with equal lengths of a diagonal is a rectangle.
A Parallelogram whose Diagonals are of Equal Length is a Rectangle
Theorem:
Prove that a parallelogram whose diagonals are of equal length is a rectangle.
Given
ABCD is a parallelogram in which AB ∥ DC, AD ∥ BC, and AC = BD.
To prove that:
ABCD is a parallelogram, i.e., in the parallelogram ABCD, one angle, say ∠BAD = 90°.
Proof:
In ∆ABC and ∆BDA,
∠CAB = ∠ACD (Since, DC ∥ AB),
∠BCA = ∠DAC(Since, AD ∥ BC),
AC = AC
Therefore, ∆ABC ≅ ∆CDA, (By AAS criterion of congruence)
Therefore, BC = AD (CPCTC).
In ∆ABC and ∆BAD,
BC = AD
AC = BD (Given),
AC = AC
Therefore, ∆ABC ≅ ∆BAD (By SSS criterion of congruence)
∠ABC = ∠BAD (CPCTC).
But ∠ABC + ∠BAD= 180° (Since, QR ∥ PS)
Therefore, ∠ABC = ∠BAD = 90°
Hence proved.
Also, See:
- Opposite Angles of a Parallelogram are Equal
- Pair of Opposite Sides of a Parallelogram are Equal and Parallel
- Opposite Sides of a Parallelogram are Equal
FAQs on a Parallelogram Whose Diagonals are of Equal Length is a Rectangle
1. Is it true that a parallelogram is a rectangle?
A parallelogram has two sets of parallel sides and two pairs of opposite sides that are congruent. A rectangle is always a parallelogram.
2. Do diagonals have an equal length?
Square, Rectangle, Parallelogram have diagonals of equal length.
3. Are the diagonals of a rectangle are equal Why?
A rectangle is a parallelogram where each angle is a right angle.
Therefore, the Diagonals of a rectangle are equal.