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Chapter 5: Q35 (page 188)

Prove: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Short Answer

Expert verified

A parallelogram is a rectangle if the diagonals of the parallelogram are congruent.

Step by step solution

01

Step 1. Consider the diagram.

The figure is made having diagonals AC and BD

Here, ABCD is a parallelogram and AC→≅BD→

02

Step 2. State the proof.

In ΔABCand ΔABD.

AB=AB (Common)

AC=BD (Given)

BC=AD (Sides of a parallelogram)

So, ΔABC≅ΔABD by SSS postulate.

Therefore, by CPCT ∠ABC≅∠BAD.

Using the property of co-interior angles in a parallelogram.

∠ABC+∠BAD=1802∠ABC=180As∠ABC≅∠BAD∠ABC=90

Therefore, parallelogram ABCD is a rectangle.

03

Step 3. State the conclusion.

Therefore, parallelogram ABCD is a rectangle if the diagonals of the parallelogram are congruent (proved).

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