According to angle sum theorem, the **sum of all interior angles** of triangle is **equal to 180 degree**.

The concept is also known as **Triangle’s Interior Angle Theorem**

Let us understand the concept with below example.

Given above is triangle ABC, the values of angles are:

∠A = 46 degree

∠B=110 degree

∠C =24 degree

According to the Triangle’s Internal Angle Theorem**∠A + ∠B +∠C = 180**

Let’s put the angle values and check if the condition satisfies or not

⟹ 46 + 110 + 24

⟹ 180

Hence, the sum of angle of triangle is 180 degree

**Note**: All triangles types; Isosceles, Scalene, Equilateral, Right triangle etc. follow this property.

If sum of internal angles does not add to 180 degrees, it means there is something wrong with the angle measurement

**Proof of Triangle Sum Theorem**

Given below is triangle ABC with interior angles ∠1, ∠2 and ∠3. Line MN is drawn such that it touched vertex A of the triangle.

**To prove**

∠1 + ∠2 + ∠3 = 180

**Solution**

Observe that line MAN is a straight line so the sum of angles will be 180 degree

∠4 + ∠1 +∠5 = 180 – – – -eq(1)

Note that line MN and BC are parallel lines

∠2 = ∠4 {alternate interior angles}

∠3 = ∠5

Putting both the values in eq(1), we get

∠2 + ∠1 +∠3 = 180

∠1 + ∠2 +∠3 = 180

Hence, we proved that sum of all interior angle of triangle measures 180 degree.

I hope you understood the above concept. Let us solve some questions for better understanding of the theorem.

**Example 01**

Given right triangle ABC with following angle measurement:

\angle A=90\ degree

\angle B=45\ degree

Find the value of angle C.

**Solution**

According to Triangle sum theorem;

∠A + ∠B +∠C = 180

Putting the value of all angles we get:

45 + 90 + ∠C = 180

∠C = 180 – 135

∠C = 45 degree

**Example 02**

Below is the triangle ABC with following angle measurement

\angle A=31\ degree

\angle C=23\ degree

Find the value of angle B and the type of triangle.

**Solution**

The triangle sum theorem states that:

∠A + ∠B +∠C = 180

Putting the values:

31 + ∠B + 23 = 180

∠B = 180 – 54

∠B = 126 degree

The figure is of Obtuse triangle.

**Example 03**

Given the triangle ABC with ∠B = 68 degree and ∠ACD = 134 degree

Find the value of angle C and angle A

**Solution**

Observe BCD is a straight line.

We know that sum of angle in a straight line is 180 degree

∠x + 134 = 180

∠x = 46 degree

We know that sum of internal angle of triangle is 180 degree

∠x + ∠y +68 = 180

46 + ∠y + 68 = 180

114 + ∠y = 180

∠y = 180 – 114

∠y = 66 degree

**Example 04**

Find the angle of triangle whose first angle is 55 degree and second angle exceeds the third angle by 10 degree.

**Solution:**

∠A = 55 degree

∠C = x degree

∠B = (x + 10) degree

We know that sum of angle of triangle = 180 degree

∠A + ∠B +∠C = 180

Putting the values

55 + (x + 10) + x = 180

65 + 2x = 180

2x = 115

x = 57.5 degree

The value of rest of the angles are:

∠C = 57.5 degree

∠B = 57.5 + 10 = 67.5 degree

**Example 05**

Find the angle measurement, if the values of three angles are:

∠A = x degree

∠B = x + 50 degree

∠C = 3x + 35 degree

**Solution**

We know that sum of internal angles of triangle is 180 degree

∠A + ∠B + ∠C =180

Putting the values, we get;

x + x + 50 + 3x + 35 = 180

5x + 85 = 180

5x = 95

x = 19

Hence the three angles are:

∠A = 19 degree

∠B = 19 + 50 = 69 degree

∠C = 3x + 35 = 92 degree

**Example 06**

ABC is a triangle in which:

∠B = 38 degree

∠C = 58 degree

Find the value of angle x and angle y

**Solution**

Consider triangle ABC;

Using angle sum property of triangle;

y + ∠B + ∠C =180

y + 38 + 58 = 180

y = 180 – 96

y = 84 degree

Observe that BAM is a straight line.

We know that sum of angle in a straight line equals 180 degree

y + x = 180

84 + x = 180

x = 180 – 84

x = 96 degree

**Example 07**

Given triangle ABC in which;

∠B = 48 degree

and side AC = BC = CD

**Solution**

Taking triangle ABC first;

Its given that side AC = BC.

We know that in triangle, equal sides have equal opposite angles.

Hence, x = y

We know that sum of internal angle = 180 degree

48 + x + y = 180

48 + x + x = 180

2x =132

x = 66

So the rest of the angles x & y = 66 degree

Observe that BCD is a straight line.

we know that Angle in straight line add up to 180 degree

So, 66 + z = 180

z = 180 – 66

z = 114 degree

In triangle ACD, its given that side AC = CD

We know that in triangle equal sides have equal opposite angles

∠CAD = ∠CDA = k

Applying sum of angle property in triangle ACD

z + k + k = 180

114 + 2k = 180

2k = 180 – 114

2k = 66

k = 33

**hence, value of angle k is 33 degrees**