Angle Sum Property of a Triangle: Definition, Theorem, Examples

What is the Angle Sum Property of a Triangle?

The “angle sum property of a triangle theorem” (also known as the “triangle sum theorem” or “angle sum theorem”) states that the sum of the three interior angles of any triangle is always $180^{\circ}$.

What is the angle sum theorem in geometry? In Euclidean geometry, any triangle whether it is a right triangle, an obtuse triangle, or an acute triangle all have interior angles that add up to 180 degrees.

Angle Sum Property of a Triangle Theorem

The angle sum property of a triangle theorem states that the sum of all three internal angles of a triangle is $180^{\circ}$. It is also known as the angle sum theorem or triangle sum theorem.

According to the angle sum theorem, in the above ABC, 

$m\angle A + m\angle B + m\angle C = 180^{\circ}$

Example: In PQR, $\angle P = 60^{\circ}, \angle Q = 70^{\circ}$

According to the angle sum theorem, in the above triangle PQR, 

$m\angle P + m\angle Q + m\angle R = 180^{\circ}$

$60^{\circ} + 70^{\circ} + m\angle R = 180^{\circ}$

$130^{\circ} + m\angle R = 180^{\circ}$

 $m\angle R = 50^{\circ}$

Triangle Sum Theorem Proof

In the figure given below, AB, BC, and CA represent three sides of triangle ABC. A, B, and C are the three vertices. $\angle A,\; \angle B,$ and $\angle C$ are the three interior angles of $\Delta ABC$.

In ∆ABC, we have to prove that the sum of the angles $\angle A,\; \angle B,$ and $\angle C$ is $180^{\circ}$. 

To prove: $m\angle A + m \angle B + m\angle C = 180^{\circ}$

Construction: Draw a line DE passing through the vertex B, which is parallel to the side AC.

At point B, two angles are formed, $\angle 1$ and $\angle 2$.

Since AB is a transversal for the parallel lines DE and AC, we have 

$m\angle 1 = m\angle A$ (since the pair of alternate interior angles are equal)

Similarly, $m\angle 2 = m\angle C$.

Now, $m\angle 1,\; m\angle B,$ and $m\angle 2$ must add up to $180^{\circ}$ (angles on a straight line). 

Thus, $\angle 1 + \angle B + \angle 2 = 180^{\circ}$… (I)

Since $\angle 1 = \angle A$ and $\angle 2 = \angle C$. … (II)

Substituting equation (II) in equation (I), we get

$m\angle A + m\angle B + m\angle C = 180^{\circ}$

Therefore, the sum of the three angles $\angle A,\; \angle B,$ and $\angle C$ is $180^{\circ}$. 

Hence, the triangle sum theorem was proved.

Exterior Angle Theorem

The exterior angle theorem states that “an exterior angle of a triangle is equal to the sum of its two opposite interior angles.”

In the above triangle, $\angle A,\; \angle B,$ and $\angle C$ are the interior angles of the triangle ABC, and $\angle 1,\; \angle 2,$ and $\angle 3$ are the exterior angle.

$m\angle A + m\angle B + m\angle C = 180^{\circ}$ (angle sum property) … (I)

Also, $m\angle A + m\angle 1 = 180^{\circ}$ (linear pair angle) … (II)

From (I) and (II), we get

$m\angle A + m\angle 1 = m\angle A + m\angle B + m\angle C$

$m\angle 1 = m\angle B + m\angle C$ 

Similarly, we can derive for other two exterior angles,∠2 and ∠3 which is given by:

$m\angle 2 = m\angle A + m\angle C$

$m\angle 3 = m\angle A + m\angle B$ 

In summary: 

Facts about Angle Sum Property of a Triangle

• The theorem of angle sum property of triangles holds true for all types of triangles.
• The sum of all exterior angles of a triangle is equal to $360^{\circ}$.
• The sum of the lengths of any two sides of a triangle is always greater than the third side.
• A rectangle can be divided into two right triangles by drawing a line from one corner to the opposite corner.
• The study of the relationship between the sides and angles of triangles is known as trigonometry.
• Due to their high strength, triangle shapes are frequently utilized in construction.
• In North Carolina, there are three cities: Raleigh, Durham, and Chapel Hill, that are often referred to as the triangle.
• The sum of all exterior angles of a triangle is equal to $360^{\circ}$.

Conclusion

In this article, we have learned all about the angle sum property of a triangle, exterior angle property of a triangle, proof of triangle sum theorem, and some important facts about triangles.

Let’s solve a few triangle angle sum theorem examples and practice problems.

Solved Examples on Angle Sum Property of a Triangle

1. In a triangle, ABC, if $m\angle A = 60^{\circ},\; m\angle B = 40^{\circ}$, then find the measure of angle $\angle C$.

Solution: 

In $\Delta ABC,\; \angle A = 60^{\circ}$ and $\angle B = 40^{\circ}$

We know that the sum of angles in a triangle is $180^{\circ}$.

$\Rightarrow  m \angle A + \angle B + \angle C = 180^{\circ}$

$\Rightarrow  60^{\circ} + 40^{\circ} + \angle C = 180^{\circ}$

$\Rightarrow  \angle C = 180^{\circ} \;−\; ( 60^{\circ} + 40^{\circ})$

$\Rightarrow  \angle C = 180^{\circ} \;−\; 100^{\circ}$

$\therefore  \angle C = 80^{\circ}$

2. One of the acute angles in a right-angled triangle is $40^{\circ}$. Using the angle sum theorem, determine the other angle. 

Solution: 

Let $\Delta ABC$ be given a right-angled triangle which is right-angled at B.

$\therefore  \angle B = 90^{\circ}$

$m\angle A = 40^{\circ}$ and we have to find out $m\angle C$.

We know that the sum of angles in a triangle is $180^{\circ}$.

$\Rightarrow  m \angle A + m \angle B + m \angle C = 180^{\circ}$

$\Rightarrow  40^{\circ} +  90^{\circ} + m\angle C = 180^{\circ}$

$\Rightarrow  m \angle C = 180^{\circ} \;−\; (40^{\circ} + 90^{\circ})$

$\Rightarrow  m \angle C = 180^{\circ} \;−\; 130^{\circ}$

$\therefore  m \angle C = 50^{\circ}$

$\therefore m \angle A = m \angle C = 50^{\circ}$

3. The measures of interior angles of a triangle are $(2x\;−\;20)^{\circ}, (3x\;−\;10)^{\circ}$^, and $2x^{\circ}$, find the values of all three angles of the triangle.

Solution: 

We know that the sum of angles in a triangle is $180^{\circ}$

$\Rightarrow (2x\;−\;20)^{\circ} + (3x\;−\;10)^{\circ} + 2x^{\circ} = 180^{\circ}$

$\Rightarrow (2x\;−\;20 + 3x\;−\;10 + 2x)^{\circ} = 180^{\circ}$

$\Rightarrow 7x\;−\;30 = 180$

$\Rightarrow 7x = 180 + 30$

$\Rightarrow 7x = 210$

$\Rightarrow x = 2107= 30$

$\Rightarrow$ Angles are $40^{\circ},\; 80^{\circ}$ and $60^{\circ}$.

4. Is it possible to construct a triangle with internal angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$?

Solution: 

Given, measurements of angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$.

Here, $43^{\circ} + 49^{\circ} + 91^{\circ} = 183^{\circ} 180^{\circ}$

We know, that, the sum of angles in a triangle is $180^{\circ}$

Hence, it is not possible to construct a triangle with measurements of angles $43^{\circ},\; 49^{\circ},$ and $91^{\circ}$.

5. In the figure given below, determine the value of “x.”

Solution: 

In the above figure, $\angle x$ is an exterior angle and $\angle A = 55^{\circ}$ and $\angle B = 47^{\circ}$ are given as interior angles.

According to the exterior angle property,

an exterior angle of a triangle is equal to the sum of its two interior opposite angles.

So, $\angle x = \angle A + \angle B = 55^{\circ} + 47^{\circ} = 102^{\circ}$

Hence, $x = 102^{\circ}$

Practice Problems on Angle Sum Property of a Triangle

Frequently Asked Questions on Angle Sum Property of a Triangle