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Chapter 5: Q35 (page 194)

Prove that if the diagonals of parallelograms are congruent, then the parallelogram is a rectangle.

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Check the figure.

Consider the figure.

02

Step 2. Step description.

The CPCTC theorem states that when two triangles are congruent, then every corresponding part of one triangle is congruent to the other.

03

Step 3. Step description.

From the figure, 螖ABCand 螖ABD.

AB=AB (Sides are common)

As the diagonals are congruent such that AC=BDand the diagonals of a parallelogram bisect such that BC=AD.

Therefore, by side-side-side postulate 螖ABC鈮�螖BAD.

Now, by CPCT theorem 鈭�ABC鈮�鈭�BAD.

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

鈭�ABC+鈭�BAD=1802鈭�ABC=180鈥夆赌夆赌�鈭�鈭�ABC鈮�鈭�BAD鈭�ABC=90

Therefore, the parallelogram is a rectangle.

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