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    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.

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    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

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    स्रोत : byjus.com

    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

    ∴∠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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    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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