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
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