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

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### Mohammed

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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 rectangleCheck out the video giveb below to know more about area of rectangle Articles to explore

1. The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y=12−x2such that the rectangle lies inside the parabola, is:-

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

4 (18)

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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. Question

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

Given: In parallelogram ABCD, AC=BD

Medium 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

Again AD ∥ ∣ Opposite sides of parallelogram ABCD

AD ∥BC and the traversal AB intersects them.

...(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 314 34

स्रोत : www.toppr.com

## 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 8Video 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.

स्रोत : www.cuemath.com

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Mohammed 14 day ago

Guys, does anyone know the answer?