# Triangle Sum Theorem

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