Famous geometry theorems.

              

  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

If you love my article and my blog then please support me and share it.

            

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

Share this post:-

Please give your valueable feedback:--

Previous

Next