# Angle sum property of triangle

Angle sum property of triangle is one of the most important concepts in triangles. The Angle sum property of triangle says that for any given triangle the sum of all the three angles is equal to ${ 180 }^{ 0 }$. We can prove Angle sum property of triangle in so many different ways. In that, I will discuss some important and easy proofs. Out of these proofs whatever you are feeling easy that proof you can learn. But if you learn all different kinds of proofs it will be good.

## One of the Proof of Angle sum property of triangle: We know that the exterior angle of any given triangle is equal to the sum of opposite interior angles of a triangle.

∠ACD = ∠A + ∠B

Now consider the straight line BCD. We know that the total sum of angles on a straight line is equal to ${ 180 }^{ 0 }$.

∠BCA + ∠ACD = ${ 180 }^{ 0 }$.

Therefore, ∠C + ∠A + ∠B = ${ 180 }^{ 0 }$.

In order, if you write then we will get ∠A + ∠B + ∠C = ${ 180 }^{ 0 }$.

Hence it is proved.

### The Practical proof of Angle sum property of triangle:  All the three angles of a triangle are arranged as shown in the above diagram. We know that angles on a straight line are equal to ${ 180 }^{ 0 }$.

∠1 + ∠2 + ∠3 = ${ 180 }^{ 0 }$.

Hence it is proved.

#### Another Proof of Angle sum property of triangle: For a triangle ABC, through the vertex A draw a line parallel to BC.

BC is parallel to EF and AC as transversal

∠ACB = ∠FAC

BC is parallel to EF and AB as transversal

∠ABC = ∠EAB

Therefore, as EF is a straight line

∠EAB + ∠A + ∠FAC = ${ 180 }^{ 0 }$.

∠ABC + ∠A + ∠ACB = ${ 180 }^{ 0 }$.

∠B + ∠A + ∠C = ${ 180 }^{ 0 }$.

Therefore, ∠A + ∠B + ∠C = ${ 180 }^{ 0 }$.

Hence it is proved. 