Question

the sum of the exterior angles of any polygon is always equal to 360
exterior angles: non - regular polygons
find the value of x.

1. x=
2. x=
3. x=
4. x=

exterior angles: regular polygons
find the measure of one exterior angle for the regular polygon with n sides.

1. quadrilateral
2. octagon
3. 16 - gon

Explanation:

1)

Step1: Use the sum of exterior angles formula

The sum of exterior angles of a polygon is \(360^{\circ}\). Let the exterior angles be \(75^{\circ},44^{\circ},65^{\circ},54^{\circ},76^{\circ},x^{\circ}\).

$$75 + 44+65 + 54+76+x=360$$

Step2: Simplify the left - hand side

$$ (75 + 44)+(65 + 54)+(76+x)=360$$

$$119+119+(76+x)=360$$

$$238+(76+x)=360$$

$$x=360-(238 + 76)$$

$$x = 46$$

2)

Step1: Use the sum of exterior angles formula

The sum of exterior angles of a polygon is \(360^{\circ}\). Let the exterior angles be \(85^{\circ},71^{\circ},48^{\circ},2x^{\circ},3x^{\circ}\).

$$85+71 + 48+2x+3x=360$$

Step2: Combine like terms

$$204 + 5x=360$$

$$5x=360 - 204$$

$$5x=156$$

$$x = 31.2$$

3)

Step1: Use the sum of exterior angles formula

The sum of exterior angles of a polygon is \(360^{\circ}\). Let the exterior angles be \(60^{\circ},80^{\circ},110^{\circ},70^{\circ},x^{\circ}\).

$$60+80 + 110+70+x=360$$

Step2: Simplify the left - hand side

$$320+x=360$$

$$x=360 - 320$$

$$x = 40$$

4)

Step1: Use the sum of exterior angles formula

The sum of exterior angles of a polygon is \(360^{\circ}\). Let the exterior angles be \(110^{\circ},70^{\circ},70^{\circ},x^{\circ}\).

$$110+70 + 70+x=360$$

Step2: Simplify the left - hand side

$$250+x=360$$

$$x=360 - 250$$

$$x = 110$$

9)

Step1: Use the formula for exterior angle of a regular polygon

The formula for the measure of an exterior angle of a regular polygon is \(\frac{360^{\circ}}{n}\). For a quadrilateral \(n = 4\).

$$ \text{Exterior angle}=\frac{360^{\circ}}{4}=90^{\circ}$$

10)

Step1: Use the formula for exterior angle of a regular polygon

For an octagon \(n = 8\).

$$ \text{Exterior angle}=\frac{360^{\circ}}{8}=45^{\circ}$$

11)

Step1: Use the formula for exterior angle of a regular polygon

For a \(16 -\)gon \(n = 16\).

$$ \text{Exterior angle}=\frac{360^{\circ}}{16}=22.5^{\circ}$$

Answer:

1. \(x = 46\)
2. \(x = 31.2\)
3. \(x = 40\)
4. \(x = 110\)
5. \(90^{\circ}\)
6. \(45^{\circ}\)
7. \(22.5^{\circ}\)