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 .
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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