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. 9) quadrilateral 10) octagon 11) 16 - go

Explanation:

Step1: Recall the sum of exterior angles

The sum of exterior angles of any polygon is \(360^{\circ}\).

Step2: Solve for \(x\) in the first non - regular polygon

For the first polygon:

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

$$314 + x=360$$

$$x = 360-314$$

$$x = 46$$

Step3: Solve for \(x\) in the second non - regular polygon

For the second polygon:

$$71+85 + 44+3x+2x=360$$

$$200+5x = 360$$

$$5x=360 - 200$$

$$5x=160$$

$$x = 32$$

Step4: Solve for \(x\) in the third non - regular polygon

For the third polygon:

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

$$320+x=360$$

$$x=360 - 320$$

$$x = 40$$

Step5: Solve for \(x\) in the fourth non - regular polygon

For the fourth polygon:

$$120+70+70+x=360$$

$$260+x=360$$

$$x=360 - 260$$

$$x = 100$$

Step6: Find the measure of one exterior angle for a quadrilateral

For a quadrilateral (\(n = 4\)):
The measure of one exterior angle \(=\frac{360}{n}=\frac{360}{4}=90^{\circ}\)

Step7: Find the measure of one exterior angle for an octagon

For an octagon (\(n = 8\)):
The measure of one exterior angle \(=\frac{360}{n}=\frac{360}{8}=45^{\circ}\)

Answer:

1. \(x = 46\)
2. \(x = 32\)
3. \(x = 40\)
4. \(x = 100\)
5. \(90^{\circ}\)
6. \(45^{\circ}\)