# if the diagonal of a parallelogram are equal then show that it is a rectangle

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## If the diagonals of a parallelogram are equal, then show that it is a rectangle.

If the diagonals of a parallelogram are equal, then show that it is a rectangle. - Get the answer to this question and access a vast question bank that is tailored for students.

Academic QuestionsMaths QuestionsIf The Diagonals Of A Parallelogram Are Equal Then Show That It Is A Rectangle

## If the diagonals of a parallelogram are equal, then show that it is a rectangle.

**Given:**

AC = BD

**To Prove:**

ABCD is a rectangle if the diagonals of a parallelogram are equal

**Solution:**

As mentioned parallelogram, so let ABCD be a parallelogram

where, given AC=BD In ∆DCB and ∆CBA,

DB = CA …………………[Given]

DC = BA …………………[Opposite sides of a parallelogram]

CB = BC …………………[Common side]

∴ ∆DCB ≅ ∆CBA [By AAA congruency]

so, ∠DCB = ∠CBA [By C.D.C.T.] ………………………….(1)

Now, DC || BA and CB is a transversal. …………………….[DCBA is a parallelogram]

∴ ∠DCB + ∠CBA = 180° [Co-interior angles of parallelogram]……………………… (2)

From (1) and (2), we have

∠DCB = ∠CBA = 90°

i.e., ABCD is a parallelogram having an angle equal to 90°.

Hence, ABCD is a rectangle. (having all angles equal to 90° and opposite sides are equal)

Thus, Proved

**Check out the video given below to know more about rectangle**

**Check out the video giveb below to know more about area of rectangle**

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2. Prove that Sum of any two sides of a triangle is greater than the third side.

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## If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Click here👆to get an answer to your question ✍️ If the diagonals of a parallelogram are equal, then show that it is a rectangle.

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## If the diagonals of a parallelogram are equal, then show that it is a rectangle.

Given: In parallelogram ABCD, AC=BDMedium Open in App Solution Verified by Toppr

To prove : Parallelogram ABCD is rectangle.

Proof : in △ACB and △BDA

AC=BD ∣ Given AB=BA ∣ Common

BC=AD ∣ Opposite sides of the parallelogram ABCD

△ACB ≅△BDA∣SSS Rule

∴∠ABC=∠BAD...(1) CPCT

Again AD ∥ ∣ Opposite sides of parallelogram ABCD

AD ∥BC and the traversal AB intersects them.

∴∠BAD+∠ABC=180 ∘

...(2) _ Sum of consecutive interior angles on the same side of the transversal is

180 ∘ From (1) and (2) , ∠BAD=∠ABC=90 ∘ ∴∠A=90 ∘ and ∠C=90 ∘

Parallelogram ABCD is a rectangle.

Video Explanation

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## If the diagonals of a parallelogram are equal, then show that it is a rectangle

If the diagonals of a parallelogram are equal, then show that it is a rectangle. We will prove that one of its interior angles is 90° and this can be done by showing two triangles congruent.

## If the diagonals of a parallelogram are equal, then show that it is a rectangle

**Solution:**

Given: The diagonals of a parallelogram are equal.

To show that a given parallelogram is a rectangle, we have to prove that one of its interior angles is 90° and this can be done by the concept of congruent triangles.

Let ABCD be a parallelogram. To show that ABCD is a rectangle, we have to prove that one of its interior angles is 90°.

In ∆ABC and ∆DCB,

AB = DC (Opposite sides of a parallelogram are equal)

BC = BC (Common)

AC = DB (Given the diagonals are equal)

∴ ∆ABC ≅ ∆DCB (By SSS Congruence rule)

⇒ ∠ABC = ∠DCB (By CPCT) ------------- (1)

It is known that the sum of the measures of angles on the same side of transversal is 180° (co - interior angles)

∠ABC + ∠DCB = 180° (AB || CD)

⇒∠ABC + ∠ABC = 180° [From equation(1)]

⇒ 2∠ABC = 180° ⇒∠ABC = 90°

Thus, ∠DCB = 90° [From equation (1)]

Hence, ∠B = ∠D = ∠C = ∠A = 90° [Since opposite angles of a parallelogram are equal].

Since ABCD is a parallelogram and one of its interior angles is 90°, ABCD is a rectangle.

**☛ Check:**NCERT Solutions for Class 9 Maths Chapter 8

**Video Solution:**

## If the diagonals of a parallelogram are equal, then show that it is a rectangle

NCERT Maths Solutions Class 9 Chapter 8 Exercise 8.1 Question 2

**Summary:**

If the diagonals of a parallelogram are equal, we have proved that it is a rectangle.

**☛ Related Questions:**

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Show that the diagonals of a square are equal and bisect each other at right angles.

Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus.

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