Educational worksheet for identifying regular and irregular polygons, with interactive exercises and a question about isosceles triangles.

Let's solve this step-by-step based on the worksheet.

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Objective:

Distinguish between regular and irregular polygons by reasoning about equal sides and angles.

> A regular polygon has:
> - All sides equal in length
> - All interior angles equal

> An irregular polygon has:
> - At least one side or angle that is different from the others

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## 🔹 Step 1: Shade the regular shapes and circle the irregular shapes (Top Section)

We are given 8 shapes:

1. Pink octagon – 8 sides, appears symmetrical → Regular ✔
➤ Shade it

2. Green star (6-pointed) – This is a star polygon, but not all sides/angles are equal unless specified. However, if it's a regular star polygon like {6/2}, it may be regular. But visually, it looks like a regular hexagram (two overlapping equilateral triangles).
➤ Likely Regular ✔
➤ Shade it

3. Yellow square – 4 equal sides, 4 equal angles → Regular ✔
➤ Shade it

4. Orange L-shape – Not all sides equal, not closed symmetrically → Irregular ✘
➤ Circle it

5. Blue right triangle – Two sides appear unequal, and angles differ → Irregular ✘
➤ Circle it

6. Yellow pentagon – Looks like a regular pentagon (all sides and angles equal) → Regular ✔
➤ Shade it

7. Blue trapezoid – Only two sides parallel, likely unequal sides → Irregular ✘
➤ Circle it

8. Black triangle – Appears to have three equal sides? Wait — it looks equilateral?
But the shape is drawn with no indication of equality. However, if it's equilateral, it’s regular. But without markings, we must judge visually. It seems equilateral → Regular ✔
➤ Shade it

> So, shade: pink octagon, green star, yellow square, yellow pentagon, black triangle
> Circle: orange L-shape, blue right triangle, blue trapezoid

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## 🔹 Step 2: Fill in the table

We have two columns: Regular and Irregular. Put a tick (√) in the correct box.

Left Table:

| Shape | Regular | Irregular |
|------|--------|----------|
| 🟨 Rectangle | | √ |
| 🟩 Hexagon | √ | |
| 🟦 Parallelogram | | √ |
| 🟡 Star (5-pointed) | √ | |
| ⬛ Square | √ | |

#### Explanation:

- Rectangle → Opposite sides equal, but adjacent sides differ → Irregular
- Hexagon → Looks like a regular hexagon (6 equal sides) → Regular
- Parallelogram → Sides not equal unless rhombus → Irregular
- Star (5-pointed) → Regular star polygon (like {5/2}) → Regular
- Square → All sides and angles equal → Regular

✔ So ticks go in:
- Rectangle → Irregular
- Hexagon → Regular
- Parallelogram → Irregular
- Star → Regular
- Square → Regular

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Right Table:

| Shape | Regular | Irregular |
|------|--------|----------|
| 🟩 Cross (plus sign) | | √ |
| 🟫 Irregular quadrilateral | | √ |
| 🟪 Triangle (pink) | √ | |
| 🟨 Star (spiky) | √ | |
| 🟣 Pentagon | √ | |

#### Explanation:

- Cross (plus sign) → Not a polygon (has intersecting lines), or if considered as a compound shape, it's not a simple polygon. Even if it were, it has unequal sides and angles → Irregular
- Brown shape → Irregular quadrilateral → Irregular
- Pink triangle → Equilateral? Yes, it looks like an equilateral triangle → Regular
- Spiky star → Looks like a regular star (e.g., {8/3} or similar) → Regular
- Purple pentagon → Looks like a regular pentagon → Regular

✔ Ticks:
- Cross → Irregular
- Brown shape → Irregular
- Pink triangle → Regular
- Spiky star → Regular
- Purple pentagon → Regular

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## 🔹 Final Question:

> This is an isosceles triangle. Is it a regular polygon? Why or why not?

 *(imaginary image)*

An isosceles triangle has:
- Two equal sides
- Two equal angles
- One side and one angle different

A regular polygon requires all sides and all angles equal.

Since an isosceles triangle does not have all three sides equal, it is not a regular polygon.

> ✔ Answer:
> No, it is not a regular polygon because only two sides are equal. For a polygon to be regular, all sides and all angles must be equal.

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✔ Summary of Answers:

#### Top Section:
- Shade (Regular):
- Pink octagon
- Green star
- Yellow square
- Yellow pentagon
- Black triangle

- Circle (Irregular):
- Orange L-shape
- Blue right triangle
- Blue trapezoid

#### Tables:

| Shapes | Regular | Irregular |
|--------|--------|----------|
| Rectangle | | √ |
| Hexagon | √ | |
| Parallelogram | | √ |
| Star (5-pt) | √ | |
| Square | √ | |

| Shapes | Regular | Irregular |
|--------|--------|----------|
| Cross | | √ |
| Brown quad | | √ |
| Pink triangle | √ | |
| Spiky star | √ | |
| Purple pentagon | √ | |

#### Final Answer:
> No, an isosceles triangle is not a regular polygon because it only has two equal sides and two equal angles, while a regular polygon must have all sides and all angles equal.

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