## 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 .