Geometry Theorems

       

 Geometry Theorems:-

Today in this article I will show you how to prove theorems related on triangles and quadrilateral.

**Prove that sum of three angles of a triangle is 180°.

Answer:- In the picture,  
To prove:- < A + < B + <C =180°
Construction:-  Through point "A"  DE∥ BC is drawn.

Proof:- ABC is a Δ , < A ,∠B and∠C are its three angles, 

∵ In Δ ABC
   DE ∥ BC and AB is the transversal,
 ∴  ∠ B = < DAB [∵ alternate angles are  equal].......(1)
 Again,
 DE ∥ BC and AC is the transversal,
∴ ∠ C = < EAC [ ∵ alternate angles are  equal].......(2)
Now (1)+(2) we get,
   < B + < C = < DAB + < EAC
again, adding < BAC in both sides of the above equation we get,
<BAC + < B + < C = < DAB +<BAC+<EAC
  
  => <BAC +<B+<C = <DAE
 => <BAC+<B+<C = 180°[<DAE is a straight angle,so it measures 180°]  
∴ <A + <B + <C = 180° Proved.

 

**Prove that sum of two remote interrior angles of a triangle is equal to its exterior angle.

Answer:-  In the picture,
ABC is a Δ, ∠A and ∠B are its two remote interrior angles,and ∠ACD is its exterrior angle.
To, prove :- <A+<B = < ACD


Construction:- Through " C" BA∥CE     is drawn.

  

Proof:- In ΔABC

        ∵ BA∥ CE and AC is the traversal,
         ∴   ∠A = <ACE [ ∵ alternate  angles are   [equal].....(1)
again , BA∥ CE and BD is the transversal,
∴ ∠ B=<ECD [ ∵ corresponding angles   are equal].....(2) 
Now, (1)+(2) we get,  
 <A + <B = < <ACE + <ECD    
=> <A + <B= <ACD    Proved.

**Prove that sum of all the angles of a quadrilateral  is 360° 

      Answer:-   
  ABCD is a quadrilateral, 
To, prove that ,<A + <B + <C +<D =360°
Construction:- Diagonal AC is joined.
         Proof:-  
       ∵ In Δ ACD, 
         we get, 
    ∵ ∠ACD + ∠ADC + ∠CAD =180°  [∴ sum of three angles of a  Δ is 360°]....(1)
    Again,  In  Δ ABC , 
   we get,
∵ ∠ABC +<ACB + <BAC =180° [∵ sum of three angles of a Δ is 180°].....(2) 
Now,  (1)+(2) we get,
∵  <CAD + <BAC)+<ABC + (<ACD+ <ACB)          <ADC = 360°
∴ < A + < B + < C + < D =360° Proved.

In the picture,

** Prove that opposite sides of a parallelogram are  equal.

Answer:-

 In the picture, 
 ABCD is a parallelogram,It has four sides AB,BC,CD and AD
    To, prove that  AB= CD  and BC = AD
Construction :-Diagonal  AC is joined. 
Proof:- ∵ In between Δs ABC and ADC  we get,   ∵
∠BAC = <ACD [ ∵ AB∥CD and AC is the transversal,so alternate angles are equal]
∵ ∠ ACB = ∠CAD [BC∥ AD and AC is the transversal,so alternate angles are equal]
Again,  AC = AC [ Common side of the Δs]    
∴ Δ ABC ≌ Δ ADC
∴ AB = CD  [ CPCT]
and BC = AD    Proved.

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FREQUENTLY ASKED QUESTIONS:-

Write a property of the diagonals of a parallelogram.

Answer:- One property of the diagonals of a parallelogram is :- diagonals of a parallelogram bisect each -other.

What are the two conditions of a quadrilateral has to satisfy to become a parallelogram.

Answer:- The two conditions of a quadrilateral which has to satisfy to become a parallelogram are:- (i) It's one pair of opposite sides must be equal and parallel. (ii)It's diagonals must bisect each other.

What is the sum of all angles of a pentagon?

Answer:- Number of sides of a penta gon (n) =5 ∴ The sum of all angles of a pentagon = {(2n-4)×90}° ={(2×5 -4)× 90}° ={(10-4) × 90}° = ( 6 × 90)° = 540°

Write the name of the largest side of a right triangle.

Answer :-The name of the largest side of right triangle is its hypotenuse.

Is it posible to make a triangle with the sides 2cm,3cm,and 5cm?

Answer :- ∵ We know that sum of any two sides of a triangle is always greater than its third side. Now we get, 3+5 = 8 > 2 5+ 2 = 7>2 ,but 2+ 3 = 5 which is equal to 3rd side 5. So in this case it is not possible to make a triangle with the sides 2cm,3cm,and 5cm .

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