Angle sum theorem states the total degrees in a polygon on adding all the interior angles of that polygon. The geometric figure in which the angle sum theorem is most predominantly used is the triangle. A triangle is a three sided figure having three edges and three vertices. Triangle angle sum theorem states that the three interior angles of the triangle add up to 180 degrees. Angle sum theorem is used in many algebra and geometry problems. It is mostly used to find out the unknown angle in a triangle or in the exterior angle theorem or even in the polygon angle sum formula.

## Definition

The sum of the angles in a triangle is equal to $180$ degrees. This statement is known as the angle sum theorem or triangle angle sum theorem.

The interior angles of the above triangle are $x, y, z$ and the vertices of the triangle are $A, B$ and $C$. The sum of the interior angles of the above triangle $ABC$ is $180$ degrees. This is called the angle sum theorem. So, as per the angle sum theorem angle $x$ + angle $y$ + angle $z$ = $180$ degrees.

## Proof

Given: Triangle $ABC$ have interior angles $x, y, z$ and line $p$ is parallel to line $q$

Required to prove: $\angle x\ +\ \angle y\ +\ \angle z$ = $180$ degrees

Statement Reasons
Triangle $ABC$ with interior angles $x, y$ and $z$ Given
$\angle m,\ \angle x$ and $\angle n$ all lies on the same straight line $p$

and at the point $A$, thus forms straight angle
By construction
$\angle m\ +\ \angle x\ +\ \angle n$ = $180$ degrees By straight angle definition
Transversal line $AC$ lies between the parallel lines $p$ and $q$ By definition of transversal
$\angle m$ and $\angle y$ forms alternate interior angle By definition of alternate interior angles
Line $p$ parallel to line $q$ By construction
$\angle m$ congruent $\angle y$ If parallel, transversal, then congruent
Transversal line $AB$ lies between the parallel lines $p$ and $q$ By definition of transversal
$\angle n$ and $\angle z$ forms alternate interior angle By definition of alternate interior angles
$\angle n$ congruent $\angle z$ If parallel, transversal, then congruent
Measure of angle $m$ = Measure of $\angle y$

and measure of $\angle n$ = measure of $\angle z$
Definition of congruent angles
$\angle m\ +\ \angle x\ +\ \angle n$ = $180$ degrees

Thus, $\angle y\ +\ \angle x\ +\ \angle z$ = $180$ degrees
By substitution property of equality

## Corollary of Angle Sum Theorem

The corollary of the Angle Sum Theorem states that if any one of the sides of the three sides of a triangle is extended or produced then the exterior angle so formed is equal to the sum of the interior opposite angles

Required to proof:

Measure of angle $4$ = Measure of angle $1$ + Measure of angle $2$

Statement Reasons
Angle $3$ and angle $4$ are supplementary
angles
Given
Measure of angle $3$ + Measure of angle
$4$ = $180$ degrees
Definition of supplementary angles
Measure of angle $1$ + Measure of angle $2$ +
Measure of angle $3$ = $180$ degrees
Angle Sum Theorem
Measure of angle $3$ + Measure of angle $4$ =
Measure of angle $1$ + Measure of angle $2$ + Measure of angle $3$
By Substitution method
Measure of angle $4$ = Measure of angle $1$ +
Measure of angle $2$
Subtracting measure of angle 3 from both sides

Thus, exterior angle (Measure of angle $4$) = Sum of opposite interior angles (Measure of angle $1$ + Measure of angle $2$)

## Triangle Angle Sum Theorem

Triangle angle sum theorem states that the sum of interior angles in a triangle is equal to $180$ degrees. It is also known as angle sum theorem or triangle sum theorem. If the interior angles in a triangle say $a, b$ and $c$ then their sum $a + b + c$ must be equal to $180$ degrees. That is, $a + b + c$ = $180$ degrees. This theorem helps us to find out the missing angle in a triangle if any. Suppose two interior angles in a triangle are given to us $40$ degrees and $50$ degrees and we are asked to find out the third angle. Then, applying the triangle angle sum theorem, we first find out the sum of the two given interior angles, that is $40$ degrees + $50$ degrees = $90$ degrees and then subtract it from $180$ degrees to get the missing angle. $180 – 90$ = $90$ degrees. Thus the missing angle is $90$ degrees.

## Polygon Angle Sum Theorem

Polygon Angle Sum theorem can be bifurcated into Polygon Interior Angle sum Theorem and Polygon Exterior Angle Sum Theorem.

Polygon interior angle sum theorem states that if a convex polygon has n sides, then the sum of the measures of interior angles of the polygon is given by $(n – 2) \times 180$ degrees.

Polygon Exterior Angle sum theorem states that the sum of all exterior angles in a polygon, one present at each of the vertex add up to $360$ degrees.

## Examples

Example 1:

Referring to the above figure, please answer the following parts of the question:

a) Find $m \angle 3$ if $m \angle 5$ = $120$ degrees and $m \angle 4$ = $60$ degrees

b) Find $m \angle 1$ if $m \angle 5$ = $145$ degrees and $m \angle 4$ = $75$ degrees

c) Find $m \angle 2$ if $m \angle 3$ = $70$ degrees and $m \angle 4$ = $50$ degrees

Solution:

a) It is known that exterior angle is equal to the sum of interior opposite angles. So,

Exterior Angle = Sum of interior opposite angles

$m \angle 5$ = $m \angle 3\ +\ m \angle 4$

Substituting the values of $m \angle 5$ and $m \angle 4$ given in the problem, we get

$120$ = $m \angle 3\ +\ 60$

$m \angle 3$ = $120 - 60$

$m \angle 3$ = $60$ degrees

b) Given are the values of $m \angle 4$ and $m \angle 5$ to find out $m \angle 1$. Looking at the figure we see that $m \angle 1$ and $m \angle 4$ are supplementary angles. It mean $m \angle 1$ and $m \angle 4$ add up to $180$ degrees. $m \angle 5$ is an additional information given which could not be used to determine the value of $m \angle 1$. So,

$m \angle 1\ +\ m \angle 4$ = $180$ degrees

$m \angle 1\ +\ 75$ = $180$

$m \angle 1$ = $180 - 75$

$m \angle 1$ = $105$ degrees

c) Triangle angle sum theorem states that the sum of interior angles in a triangle is equal to $180$ degrees. The interior angles in the given triangle figure are $m \angle 2,\ m \angle 3$ and $m \angle 4$. So,

$m \angle 2\ +\ m \angle 3\ +\ m \angle 4$ = $180$

Substituting the values of $m \angle 2$ and $m \angle 3$ given in the problem, we get

$m \angle 2\ +\ 70\ +\ 50$ = $180$

$m \angle 2\ +\ 120$ = $180$

$m \angle 2$ = $180 - 120$

$m \angle 2$ = $60$ degrees
Example 2:

The measure of one exterior angle of a regular polygon is given. Find the number of sides for each:

a) $10$ degrees

b) $90$ degrees

Solution:

We know that, each exterior angle of a regular polygon = $\frac{360}{n}$

Where, $n$ stands for the number of sides in a polygon

Substituting the value of each exterior angle given in the problem, we get

$10$ = $\frac{360}{n}$

$N$ = $\frac{360}{10}$

$N$ = $36$

Thus, the number of sides in the polygon is $36$

b) We know that, each exterior angle of a regular polygon = $\frac{360}{n}$

Where, $n$ stands for the number of sides in a polygon

Substituting the value of each exterior angle given in the problem, we get

$90$  = $\frac{360}{n}$

$N$ = $\frac{360}{90}$

$N$ = $4$

Thus, the number of sides in the polygon is $4$
Example 3:

The measure of an exterior angle of a regular polygon is $3x$, and the measure of an interior angle is $6x$

a) Use the relationship between exterior and interior angles to find out $x$

b) Find the measure of one interior angle and one exterior angle

c) Find the number of sides in the polygon and type of polygon

Solution:

a) In a polygon, an interior angle and the corresponding exterior angle are supplementary angles. So, interior angle + exterior angle in a polygon = $180$

Substituting the values of interior angle and exterior angle given in the problem, we get

$3x + 6x$ = $180$

$9x$ = $180$

$X$ = $\frac{180}{9}$

$X$ = $20$

Thus, the value of $x$ is $20$

b) The measure of one interior angle in the polygon is given $6x$. Substituting the value of $x$ in it, we get $6x$ = $6 \times 20$ = $120$ degrees

The measure of one exterior angle in the polygon is given $3x$. Substituting the value of $x$ in it, we get $3x$ = $3 \times 20$ = $60$ degrees

c) We know that, each exterior angle of a regular polygon = $\frac{360}{n}$

Where, $n$ stands for the number of sides in a polygon

Substituting the value of each exterior angle given in the problem, we get

$60$ = $\frac{180}{9}$

$N$ = $\frac{360}{60}$

$N$ = $6$

Thus, the number of sides in the polygon is $6$ and the type of polygon is hexagon