This worksheet helps students master the properties of regular polygons by completing key formulas and calculating angles for different side counts.

Let's solve this step by step.

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Step 1: Complete the formulas

We are given a regular polygon with $ n $ sides. Let’s fill in the blanks for the formulas:

1. Sum of exterior angles = $ \boxed{360^\circ} $

> ✔ The sum of the exterior angles of any polygon is always $ 360^\circ $, regardless of the number of sides.

2. Interior angle + Exterior angle = $ \boxed{180^\circ} $

> ✔ These two angles form a straight line at each vertex.

3. Exterior angle = $ \boxed{360^\circ} \div n $

> ✔ Since the total of all exterior angles is $ 360^\circ $, and they are equal in a regular polygon, divide by $ n $.

4. Sum of interior angles = $ \boxed{180^\circ} \times (n - \boxed{2}) $

> ✔ This is the standard formula: $ (n - 2) \times 180^\circ $

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Now let’s complete the table using these formulas.

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Step 2: Fill in the Table

We’ll go row by row and use the relationships:

- Exterior angle = $ \frac{360^\circ}{n} $
- Interior angle = $ 180^\circ - \text{Exterior angle} $
- Sum of interior angles = $ (n - 2) \times 180^\circ $
- Also, Interior angle = $ \frac{\text{Sum of interior angles}}{n} $

Let’s start filling in the table.

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#### Row 1: n = 3

- Sum of interior angles = $ 180^\circ $ (given)
- So: $ (n - 2) \times 180 = (3 - 2) \times 180 = 180^\circ $ → correct
- Each interior angle = $ 180^\circ / 3 = 60^\circ $
- Exterior angle = $ 180^\circ - 60^\circ = 120^\circ $

✔ So:
- Exterior Angle = $ 120^\circ $
- Interior Angle = $ 60^\circ $
- Sum of Interior Angles = $ 180^\circ $ (already given)

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#### Row 2: n = 6

- $ n = 6 $
- Sum of interior angles = $ (6 - 2) \times 180 = 4 \times 180 = 720^\circ $
- Each interior angle = $ 720^\circ / 6 = 120^\circ $
- Exterior angle = $ 180^\circ - 120^\circ = 60^\circ $

✔ So:
- Exterior Angle = $ 60^\circ $
- Interior Angle = $ 120^\circ $
- Sum of Interior Angles = $ 720^\circ $

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#### Row 3: Exterior Angle = 18°

- $ \text{Exterior angle} = \frac{360^\circ}{n} = 18^\circ $
- So $ n = \frac{360}{18} = 20 $
- Interior angle = $ 180^\circ - 18^\circ = 162^\circ $
- Sum of interior angles = $ (20 - 2) \times 180 = 18 \times 180 = 3240^\circ $

✔ So:
- $ n = 20 $
- Exterior Angle = $ 18^\circ $
- Interior Angle = $ 162^\circ $
- Sum of Interior Angles = $ 3240^\circ $

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#### Row 4: Interior Angle = 140°

- Interior angle = $ 140^\circ $
- So exterior angle = $ 180^\circ - 140^\circ = 40^\circ $
- $ \text{Exterior angle} = \frac{360^\circ}{n} = 40^\circ $
- So $ n = \frac{360}{40} = 9 $
- Sum of interior angles = $ (9 - 2) \times 180 = 7 \times 180 = 1260^\circ $

✔ So:
- $ n = 9 $
- Exterior Angle = $ 40^\circ $
- Interior Angle = $ 140^\circ $
- Sum of Interior Angles = $ 1260^\circ $

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#### Row 5: n = 8

- $ n = 8 $
- Exterior angle = $ 360^\circ / 8 = 45^\circ $
- Interior angle = $ 180^\circ - 45^\circ = 135^\circ $
- Sum of interior angles = $ (8 - 2) \times 180 = 6 \times 180 = 1080^\circ $

✔ So:
- Exterior Angle = $ 45^\circ $
- Interior Angle = $ 135^\circ $
- Sum of Interior Angles = $ 1080^\circ $

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#### Row 6: Sum of interior angles = 540°

- $ (n - 2) \times 180 = 540 $
- Divide both sides: $ n - 2 = 540 / 180 = 3 $
- So $ n = 5 $
- Each interior angle = $ 540 / 5 = 108^\circ $
- Exterior angle = $ 180 - 108 = 72^\circ $

✔ So:
- $ n = 5 $
- Exterior Angle = $ 72^\circ $
- Interior Angle = $ 108^\circ $
- Sum of Interior Angles = $ 540^\circ $

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#### Row 7: Interior Angle = 150°

- Interior angle = $ 150^\circ $
- Exterior angle = $ 180 - 150 = 30^\circ $
- $ n = 360 / 30 = 12 $
- Sum of interior angles = $ (12 - 2) \times 180 = 10 \times 180 = 1800^\circ $

✔ So:
- $ n = 12 $
- Exterior Angle = $ 30^\circ $
- Interior Angle = $ 150^\circ $
- Sum of Interior Angles = $ 1800^\circ $

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#### Row 8: Sum of interior angles = 360°

- $ (n - 2) \times 180 = 360 $
- $ n - 2 = 360 / 180 = 2 $
- $ n = 4 $
- Each interior angle = $ 360 / 4 = 90^\circ $
- Exterior angle = $ 180 - 90 = 90^\circ $

✔ So:
- $ n = 4 $
- Exterior Angle = $ 90^\circ $
- Interior Angle = $ 90^\circ $
- Sum of Interior Angles = $ 360^\circ $

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#### Row 9: Sum of interior angles = 1440°

- $ (n - 2) \times 180 = 1440 $
- $ n - 2 = 1440 / 180 = 8 $
- $ n = 10 $
- Each interior angle = $ 1440 / 10 = 144^\circ $
- Exterior angle = $ 180 - 144 = 36^\circ $

✔ So:
- $ n = 10 $
- Exterior Angle = $ 36^\circ $
- Interior Angle = $ 144^\circ $
- Sum of Interior Angles = $ 1440^\circ $

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#### Row 10: Interior Angle = 156°

- Interior angle = $ 156^\circ $
- Exterior angle = $ 180 - 156 = 24^\circ $
- $ n = 360 / 24 = 15 $
- Sum of interior angles = $ (15 - 2) \times 180 = 13 \times 180 = 2340^\circ $

✔ So:
- $ n = 15 $
- Exterior Angle = $ 24^\circ $
- Interior Angle = $ 156^\circ $
- Sum of Interior Angles = $ 2340^\circ $

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✔ Final Completed Table

| $ n $ | Exterior Angle | Interior Angle | Sum of Interior Angles |
|--------|----------------|----------------|-------------------------|
| 3 | 120° | 60° | 180° |
| 6 | 60° | 120° | 720° |
| 20 | 18° | 162° | 3240° |
| 9 | 40° | 140° | 1260° |
| 8 | 45° | 135° | 1080° |
| 5 | 72° | 108° | 540° |
| 12 | 30° | 150° | 1800° |
| 4 | 90° | 90° | 360° |
| 10 | 36° | 144° | 1440° |
| 15 | 24° | 156° | 2340° |

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✔ Final Answer: Formulas

- Sum of exterior angles = 360°
- Interior angle + Exterior angle = 180°
- Exterior angle = 360° ÷ n
- Sum of interior angles = 180° × (n – 2)

And the completed table is as above.

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