## Famous geometry theorems.

- Today in this article I will discuss on some geometry theorems of quadrilaterals for class 9 and class 10 which are very famous in mathematics education.

- Prove that if one pair of opposite side of a quadrilateral is equal and parallel then it is a parallelogram.

### Answer:-

In the picture, ABCD is a quadrilateral, whose AB ∥ CD ,also AB = CD, To prove that,ABCD is a parallelogram.

Proof:- ∵ AB∥ CD and AC is the transversal

∴ ∠BAC = <ACD [∵ alternate angles are

equal].....(1)

Now, in between Δs ABC and ACD we get,

∵ ∠BAC = <ACD [ from….(1)]

∵ AB = CD [ ∵ given]

and, AC = AC [∵ common side of the Δs]

∴ Δ ABC ≌ Δ ACD [∵by SAS congruence

rule]

∴ BC = AD [∵ by CPCT]

and <ACB = <CAD

but , <ACB and < CAD are alternate angles and AC is the transversal between them,

∴ BC ∥ AD

Now, in quadrilateral ABCD, we find,

∵ BC ∥ AD, BC= AD [∵ Proved]

and AB∥ CA, AB= CD [∵ given]

So,

Opposite sides of quadrilateral ABCD are equal and parallel.

∴ ABCD is a parallelogram , Proved.

### Construction:- Diagonal AC is joined.

- Prove that if opposite sides of a quadrilateral are equal, then it is a parallelogram

###
Answer:-

In the picture, ABCD is a quadrilateral,its
AB= CD and BC = AD To prove, ABCD is a parallelogram.

Construction:- Diagonal AC is joined.

Proof:-

∵ In between Δs ABC and ADC

we get,

∵ AB = CD [ ∵ given]

∵ BC = AD [∵ given]

and AC = AC [ ∵Common side of the Δs]

∴ Δ ABC ≌ Δ ADC [ ∵by SSS congruence rule]

∴ ∠BAC = < ADC [by CPCT]
,

but they are alternate angles and AC is the transversal between
them,
∴ AB∥CD, similarly we can prove
BC∥AD

Now in quadrilateral, ABCD

we find
AB = CD [∵given]
and BC = AD
and AB∥ CD [ ∵ proved]
also BC∥ AD

So, opposite sides of the quadrilateral
ABCD are equal and parallel

∴ ABCD is a parallelogram, proved.

- Prove that diagonals of a parallelogram bisect eachother.

###
Answer:-

In the picture, ABCD is a parallelogram, AC and BD are its diagonals intersect each other at point " O"

To prove, AO = OC
and BO = OD

Proof:-

∵ In between Δs AOB and COD

we get,

∵ AB = CD [∵ Opposite sides of the
parallelogram]

∵ ∠AOB =<COD[∵vertically opposite angles]

and,

<OAB = <OCD [∵alternate angles]

rule]

∴ AO = OC

and [ ∵ by CPCT]

BO = OD

Proved.

### ∴ Δ AOB ≌ ΔCOD [∵by ASA congruence

- Prove that diagonals of a rectangle are equal.

###
Answer:-

In the picture, PQRS is a rectangle, PR and QS are its two diagonals.

To prove:- PR = QS

Proof:- ∵ In between Δs PQR and PQS

we get,

∵ QR = PS [∵ Opposite sides of the
rectangle]

∵ PQ = PQ [∵ common side of the Δs]

and

<PQR = <QPS [∵ABCD is rectangle,so both angles are 90°]

### ∴ Δ PQR ≌ Δ PQS

### ∴ PR = QS [by CPCT]

Proved.

- Prove that if one angle of a rectangle is 90°, then all angles of the rectangle will be right angle.

### Answer:-

### Let, XYZP is a rectangle,its <X=90°

### To prove;-<X=<Y=<Z=<P=90°= right angle

### Proof:- ∵ ABCD is a rectangle,

∴∠X +∠Y=180°[∵sum of adjacent angles of a rectangle is 180°]

=> 90° + <Y = 180°[ <X=90°given]

=> <Y = 180°- 90°

=> <Y= 90°

Similarly,

we can prove, <Z =<P=90°

Now, in rectangle XYZP

<X=<Y=<Z=<P=90°= right angle ,proved

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## FREQUENTLY ASKED QUESTIONS:-
## Write two properties of parallelogram?

Answer :- Two properties of parallelogram are :-
(i) Opposite sides of the parallelogram are equal and parallel.
(ii) Diagonals of a parallelogram bisect each other at their point of contact.

## Write three properties of a rectangle.

Answer:-Three properties of a rectangle are :-
(i) Diagonals of a rectangle are equal.
(ii) Each angle of a rectangle is 90°
(iii) Opposite sides are equal and parallel.

## Write two similarities between rhombus and square.

Answer :- Two similarities between rhombus and square are :-
(i) All sides of both are equal.
(ii) Both's diagonals bisect each-other perpendicularly.

## What is the relation of areas between a triangle and a rectangle if both are standing on the same base and in between same parallels?

Answer:- The relation is,
area of triangle = ½ area of rectangle

##
Name the quadrilateral whose one pair of sides are parallel.

Answer:- Trapezium

## Write two properties of parallelogram?

Answer :- Two properties of parallelogram are :- (i) Opposite sides of the parallelogram are equal and parallel. (ii) Diagonals of a parallelogram bisect each other at their point of contact.

## Write three properties of a rectangle.

Answer:-Three properties of a rectangle are :- (i) Diagonals of a rectangle are equal. (ii) Each angle of a rectangle is 90° (iii) Opposite sides are equal and parallel.

## Write two similarities between rhombus and square.

Answer :- Two similarities between rhombus and square are :- (i) All sides of both are equal. (ii) Both's diagonals bisect each-other perpendicularly.

## What is the relation of areas between a triangle and a rectangle if both are standing on the same base and in between same parallels?

Answer:- The relation is, area of triangle = ½ area of rectangle

## Name the quadrilateral whose one pair of sides are parallel.

Answer:- Trapezium