This worksheet guides students to discover the sum of interior angles by dividing polygons into triangles.

Let's solve this step-by-step based on the worksheet titled "Investigation – Angles in polygons".

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🔷 Task 1: Draw lines from one vertex to split each polygon into triangles

You are given these polygons:
- Rectangle (quadrilateral)
- Pentagon
- Hexagon
- Heptagon
- Octagon

For each polygon, pick one vertex and draw straight lines to all other non-adjacent vertices. This will divide the polygon into triangles.

💡 Key idea: From one vertex, you can draw diagonals to all other vertices except the two adjacent ones (since those are already connected by sides). Each diagonal splits the shape into more triangles.

But since this is a visual task, we'll focus on the pattern and use it for Task 2.

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🔷 Task 2: Complete the table

We need to fill in:

| Name of polygon | Number of sides | Number of triangles inside polygon | Sum of interior angles |
|------------------|------------------|------------------------------------|------------------------|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | ? | ? |
| Pentagon | 5 | ? | ? |
| Hexagon | 6 | ? | ? |
| Heptagon | 7 | ? | ? |
| Octagon | 8 | ? | ? |

#### 🧠 Key Concept:
When you draw diagonals from one vertex of an *n*-gon, you can form (n - 2) triangles.

> ✔ Formula:
> Sum of interior angles = (n - 2) × 180°

Let’s compute each row:

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#### 🔹 Triangle (3 sides)
- Triangles: 3 - 2 = 1
- Sum: 1 × 180° = 180° ✔

#### 🔹 Quadrilateral (4 sides)
- Triangles: 4 - 2 = 2
- Sum: 2 × 180° = 360°

#### 🔹 Pentagon (5 sides)
- Triangles: 5 - 2 = 3
- Sum: 3 × 180° = 540°

#### 🔹 Hexagon (6 sides)
- Triangles: 6 - 2 = 4
- Sum: 4 × 180° = 720°

#### 🔹 Heptagon (7 sides)
- Triangles: 7 - 2 = 5
- Sum: 5 × 180° = 900°

#### 🔹 Octagon (8 sides)
- Triangles: 8 - 2 = 6
- Sum: 6 × 180° = 1080°

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✔ Completed Table:

| Name of polygon | Number of sides | Number of triangles inside polygon | Sum of interior angles |
|------------------|------------------|------------------------------------|------------------------|
| Triangle | 3 | 1 | 180° |
| Quadrilateral | 4 | 2 | 360° |
| Pentagon | 5 | 3 | 540° |
| Hexagon | 6 | 4 | 720° |
| Heptagon | 7 | 5 | 900° |
| Octagon | 8 | 6 | 1080° |

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🔷 Task 3: Questions

#### ❓ What is the difference between the interior angle sum of a triangle and a quadrilateral?

- Triangle: 180°
- Quadrilateral: 360°
- Difference: 360° − 180° = 180°

✔ Answer: 180°

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#### ❓ What is the difference between the interior angle sum of a hexagon and a heptagon?

- Hexagon: 720°
- Heptagon: 900°
- Difference: 900° − 720° = 180°

✔ Answer: 180°

💡 Notice: Every time you add a side, you add one more triangle, so the sum increases by 180°.

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#### ❓ Predict the interior angle sum of a nonagon (9 sides) and a decagon (10 sides)

Use the formula:
(n - 2) × 180°

- Nonagon (n = 9):
(9 - 2) × 180° = 7 × 180° = 1260°

- Decagon (n = 10):
(10 - 2) × 180° = 8 × 180° = 1440°

✔ Answers:
- The interior angle sum of a nonagon is 1260°
- The interior angle sum of a decagon is 1440°

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✔ Final Summary:

#### 🔸 Pattern:
- For any n-gon, the number of triangles formed from one vertex is (n - 2)
- So, sum of interior angles = (n - 2) × 180°
- Each additional side adds 180° to the total interior angle sum.

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📌 Conclusion:

This investigation shows that the sum of interior angles of a polygon increases by 180° with each additional side, because each new side allows one more triangle to be formed when dividing the polygon.

Let me know if you'd like a diagram or explanation for how to draw the diagonals!