**Angle Sum Property of a Triangle says that Sum of all the Angles of the Triangle is always equal to 180°.**

For Better understanding of Angle Sum Property, study the following examples :-

**Example 1** = Below diagram represent Triangle ABC

In the above diagram, Triangle ABC has

∠ A = 45°

∠ B = 90°

∠ C = 45°

**Now as per the Angle Sum Property,Sum of ∠ A, ∠ B & ∠ C has to be equal to 180°.**

Lets Check the property :-**∠ A + ∠ B + ∠ C = 180°**

Put the values of all angle and we get:

45° + 90° + 45° = 180°

180° = 180°

since L.H.S = R.H.S

Hence its proved that **Sum of all the Angles of the Triangle is always equal to 180°.**

**Example 2** = Below diagram represent Triangle PQR

In the above diagram, Triangle ABC has

∠ P = 70°

∠ Q = 60°

∠ R = 50°

**Now as per the Angle Sum Property,Sum of ∠ P, ∠ Q & ∠ R has to be equal to 180°.**

Lets Check the property :-**∠ P + ∠ Q + ∠ R = 180°**

Put the values of all angle and we get:

70° + 60° + 50° = 180°

180° = 180°

since L.H.S = R.H.S

Hence its proved that **Sum of all the Angles of the Triangle is always equal to 180°.**

** How to prove angle sum property of triangle:**

There are several methods to prove angles sum property of triangle, but here we have explained three methods only.

** Method 1: Prove angle sum property of triangle with the help of protractor **

Draw a triangle and measure all its angles with the help of protractor.

Now add the measurement and you will notice that sum of these measurement is equal to 180° (ignore marginal error).

You can try this with few more triangles and you will notice that in all other triangles too; sum of measurement of all the angles is equal to 180°.

Hence, this prove angles sum property of triangle, which says that sum of all the angles of triangle is equals to 180°

** Method 2: Prove angle sum property of triangle with the help of exterior angle property. **

Before you study this method you are advised to read:

What is Exterior Angle Property of a Triangle ?

What is Linear Pair ?

Draw a triangle ABC and produce BC to D as shown in the following figure:

Mark all the angles of triangles as ∠ 2, ∠ 3 & ∠ 4 as shown in the above figure.

Also, mark exterior angle as ∠ 1 as shown in the above figure.

Now as per Exterior Angle Property which says that exterior angle is equal to sum of interior opposite angles; we get:

∠ 1 = ∠ 4 + ∠ 3 .....(statement 1)

Also, ∠ 1 & ∠ 2 from linear pair, so we get:

∠ 1 + ∠ 2 = 180°

Put the Values of angles ∠ 1 from (statement 1) and we get:

∠ 4 + ∠ 3 + ∠ 2 = 180°

And ∠ 4, ∠ 3 & ∠ 2 all are the angles of triangle ABC.

Hence, Angle Sum Property is proved which says that Sum of all the angles of a triangle are equals to 180°

** Method 3: Prove angle sum property of triangle with the principles of parallel lines cut by a transversal **

Before you understand this method, you are advised to read:

What is Transversal Line ?

What are the Properties of Transversal ?

What is Linear Pair ?

In the below diagram we have a triangle ABC and we have to prove the angle sum property of triangle i.e. sum of it's angles is equal to 180°

Or we write it as:

∠ 1 + ∠ 2 + ∠ 3 = 180°

** Following are the steps to prove Angle Sum Property of Triangle: **

∠ 1 and ∠ 4 are alternative interior (as shown below)

And as per the property of transversal, ** If two parallel lines are cut by a transversal, then pair of alternative interior angles are equal**. So we get:

∠ 1 = ∠ 4 ..... (Statement 1)

∠ 2 and ∠ 5 are alternative interior angles (as shown below):

And as per the property of transversal, "If two parallel lines are cut by a transversal, then pair of alternative interior angles are equal". So we get:

∠ 2 = ∠ 5 ..... (Statement 2)

Now, you can observe that Angle 3, Angle 4 and Angle 5 lies on the line PQ, thereby forming a Linear Pair, so we get:

∠ 4 + ∠ 3 +∠ 5 = 180°

As proved in statement 1 and 2 above (∠ 4 = ∠ 1 & ∠ 5 = ∠ 2), so we get:

∠ 1 + ∠ 3 + ∠ 2 = 180°

Or we can write it as:

∠ 1 + ∠ 2 + ∠ 3 = 180°

Hence, this proved angle sum property of triangle with the principles of parallel lines cut by a transversal