**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