Question

the diagonals of a parallelogram are equal then show that it is a rectangular​

Answer 1

Given :-

A parallelogram ABCD in which the diagonals are equal i.e. AC = BD

To Prove :-

ABCD is a rectangle.

Proof :-

In ∆ ABC and ∆ ADC ,

• AB = DC ( Opposite sides of a parallelogram are equal )
• AC = BD ( Given )
• AC = CA ( Common )

Therefore, by SSS criterion for congruence ∆ ABC and ∆ ADC are congruent.

Therefore,

• angle ABC = angle DCB ( by CPCT ) { Statement 1 }

Now, since ABCD is a parallelogram,

Therefore, AB II DC ( Since, opposite sides of a parallelogram are parallel )

So, let CB be a transversal. Now,

$\sf \longrightarrow \: \angle \: B + \angle \:C = 180^\circ$ ( Since the angles on the same side of the transversal add up to 180°

And according to statement 1, angle B and angle C are equal. Therefore,

$\sf \longrightarrow \: \angle \: B + \angle \:C = 180 ^\circ \\ \sf \longrightarrow \: 2 \angle \: B = 180 ^\circ\: \: \: \: \: \: \: \: \: \\ \sf \longrightarrow\angle \: B = \frac{180^\circ}{2} = 90^\circ \: \:$

Therefore the value of one of the angles is 90 and we know that the opposite angles of a parallelogram are equal. Hence, all the angles measure 90°.

Therefore, ABCD is a rectangle.

Hence, proved.

Know More :-

There are some criterions for congruency of two triangles. They are as follows :-

• SSS criterion
• SAS criterion
• ASA criterion
• RHS criterion

1. SSS CRITERION :- ( side-side-side )

This criterion states that if two triangles have same measure of corresponding sides then they are congruent.

2. SAS CRITERION :- ( side - angle - side )

This criterion states that if two triangles have two corresponding sides and the including angle with equal measures then they are said to be congruent.

3. ASA CRITERION :- ( angle- side - angle )

This criterion states that if two triangles have two corresponding angles and including side equal then they are said to be congruent.

4. RHS CRITERION :- ( right angle - hypotenuse - side )

This criterion states that if the corresponding hypotenuse and side of two triangles measure same then the triangle is said to be congruent.