Question

for questions 1 - 3, sketch the transformation.

1. reflection across the x axis
2. rotation 90° ccw
3. translation 2 left, 3 down

for questions 4 - 6, describe the transformation from green to red. (example: translation: 3 up, 5 right)
4.
5.
6.

1. what is the formula to determine the sum of the interior angles of a polygon?
2. what is the sum of the measure of the exterior angles of any polygon?
3. what is the measure of each interior angle of a regular dodecagon (12 sides)? (show work)
4. what is the measure of the exterior angles of a regular octagon? (show work)

Explanation:

Step1: Recall the formula for the sum of interior angles

The formula for the sum of interior angles of a polygon is \(S=(n - 2)\times180^{\circ}\), where \(n\) is the number of sides of the polygon.

Step2: Recall the sum of exterior angles

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

Step3: Find the measure of each interior angle of a regular dodecagon

For a dodecagon, \(n = 12\). First, find the sum of interior angles using \(S=(n - 2)\times180^{\circ}=(12 - 2)\times180^{\circ}=10\times180^{\circ} = 1800^{\circ}\). Since it is a regular dodecagon, each interior angle \(\theta=\frac{S}{n}=\frac{1800^{\circ}}{12}=150^{\circ}\).

Step4: Find the measure of each exterior angle of a regular octagon

For an octagon, \(n = 8\). Since the sum of exterior angles is \(360^{\circ}\), and for a regular octagon, each exterior angle \(\alpha=\frac{360^{\circ}}{n}=\frac{360^{\circ}}{8} = 45^{\circ}\).

Answer:

1. \(S=(n - 2)\times180^{\circ}\)
2. \(360^{\circ}\)
3. \(150^{\circ}\)
4. \(45^{\circ}\)