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# if each pair of opposite sides of a quadrilateral is equal then it is a parallelogram

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### Mohammed

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## If each pair of opposite sides of a quadrilateral is equal the it is a parallelogram.

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Updated on : 2022-09-05

Solution Verified by Toppr

Construction:- Joi A and C.

Proof:- In △ABC and △CDA, AB=CD(Given) AD=BC(Given) AC=AC(Common) ∴△ABC≅△CDA(By SSS)

Therefore, by CPCTC,

∠ACD=∠BAC ∠DAC=∠ACB

Since alternate anglea are equal, thus the ines are parallel.

Since both the pairs of opposite sides of quadrilateral are parallel, ABCD is a parallelogram.

Hence proved.

255 13

स्रोत : www.toppr.com

## If each pair of opposite sides of a quadrilateral are equal, then it is a :

If each pair of opposite sides of a quadrilateral are equal, then it is a :

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If each pair of opposite sides of a quadrilateral are equal, then it is a :

Question

If each pair of opposite sides of a quadrilateral are equal, then it is a :

A Square B Parallelogram C Rhombus D Kite Open in App Solution

The correct option is B Parallelogram

A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel.

Let A B C D

A B = C D and A B | | C D Join A, C Now, in Δ’s A B C a n d A D C , A B = C D (Given) A C = A C (Common side) ∠ B A C = ∠ A C D (Alternate angles) Δ A B C ≅ Δ C D A

[By SAS Congruence criterion]

∠ B C A = ∠ D A C [CPCT] Hence A D | | B C

[Since alternate angles are equal, lines are parallel]

A B C D , A B | | C D a n d A D | | B C

Since both the pairs of opposite sides of quadrilateral are parallel,

A B C D is a parallelogram. Suggest Corrections 10

SIMILAR QUESTIONS

Q. If each pair of opposite sides of a quadrilateral are equal, then it is a :Q. If each pair of opposite sides of a quadrilateral are equal, then what type of quadrilateral it is?Q. If each pair of opposite sides of a quadrilateral are equal and parallel, then it is a ____________.Q.

Complete each of the following statements by means of one of those given in brackets against each :

(i) If one pair of opposite sides is equal and parallel, then the figure is ........

(parallelogram, rectangle, trapezium)

(ii) If in a quadrilateral only one pair of opposite sides is parallel, the quadrilateral is .......

(square, rectangle, trapezium)

(iii) A line is drawn from the mid-point of one side of a triangle ......, another side intersects the third side at its mid-point.

(perpendicular to, parallel to, to meet)

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a ......

(rectangle, square, rhombus)

(v) The consecutive angle of a parallelogram is......

(supplementary, complementary)

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ........

(rectangle, parallelogram, rhombus)

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a ...........

(parallelogram, rhombus, rectangle)

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a ..........

(kite, rhombus, square)

Q. Complete each of the following statements by means of one of those given in brackets against each:

(i) If one pair of opposite sides are equal and parallel, then the figure is ........................

(parallelogram, rectangle, trapezium)

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ................ (square, rectangle, trapezium)

(iii) A line drawn from the mid-point of one side of a triangle .............. another side intersects the third side at its mid-point. (perpendicular to parallel to, to meet)

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a .................

(rectangle, square, rhombus)

(v) Consecutive angles of a parallelogram are ...................

(supplementary, complementary)

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ...............

(rectangle, parallelogram, rhombus)

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a ....................

(parallelogram, rhombus, rectangle)

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a ..................

(kite, rhombus, square)

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स्रोत : byjus.com

## If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogram.Can you reason out why?

If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogram.Can you reason out why?. Ans: Hint: We know that, if two opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is known a...

If each pair of opposite sides of a quadrilateral is equal, then the quadrilateral is a parallelogram.

Can you reason out why?

Answer Verified 217.8k+ views 3 likes

Hint: We know that, if two opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is known as parallelogram.

For this proof, we will try to find the congruence of triangles.

By using the congruent property, we can make the alternate angles are equal and the opposite sides are parallel.

Complete step-by-step solution:

It is given that; each pair of opposite sides of a quadrilateral is equal.

We have to show that the quadrilateral is a parallelogram.

Let us draw the diagram.

Here, ABCD ABCD

We have to prove that

ABCD ABCD

is a parallelogram. Moreover, we have to prove

AB∥CD AB∥CD and AD∥BC AD∥BC . Let us join A A and C C . In △ABC △ABC and △CDA △CDA It is given that, AB=CD AB=CD It is given that, AD=BC AD=BC AC AC is the common side.

So, by SSS condition both the triangles are congruent.

That is, △ABC≅△CDA △ABC≅△CDA

Then, by common property of common triangles we get,

∠ACD=∠BAC ∠ACD=∠BAC ∠DAC=∠ACB ∠DAC=∠ACB

Since, the alternate angles are equal, thus the lines are parallel.

Since both the pairs of opposite sides of a quadrilateral are parallel,

ABCD ABCD is a parallelogram.

Hence, proved.Note: We have mind that, there are three condition for congruence,

SSS - The triangles are said to be congruent if all the three sides of one triangle are equal to the three corresponding sides of another triangle.

SAS - The triangles are said to be congruent if the correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle.

AAS - The triangles are said to be congruent if two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle.

स्रोत : www.vedantu.com

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Mohammed 7 day ago

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