Question

a parallelogram must be a rectangle if its diagonals

Answer 1

A parallelogram must be a rectangle if its diagonals are Congruent ( Equal in length)

Parallelogram

• Opposite sides are Parallel and Equal
• Opposite angles are equal
• Adjacent angles are supplementary

Rectangle

• A parallogram with each angle right angle ( 90°)

Let say ABCD is a parallogram

then AD = BC

Now if AC = BD  ( congruent diagonals) Equal in length

Now in ΔABD and ΔBAC

AB ≅ BA    ( Same segment)

AD ≅ BC  ( opposite side of parallelogram)

AC ≅  BD    ( Assumed)

=> ΔABD ≅ ΔBAC

=> ∠BAD ≅ ∠ABC

=> m ∠BAD = m∠ABC

m ∠BAD + m∠ABC  = 180° ( Adjacent angles of parallelogram)

Hence  m ∠BAD = m∠ABC  = 90°

Hence ABCD is a parallelogram with right angle (Requirement of Rectangle)  if  its diagonals are Congruent ( Equal in length)

A parallelogram must be a rectangle if its diagonals are Congruent ( Equal in length)