Understanding Quadrilaterals
Polygons & Their Angles
What is a Polygon? A polygon is a closed two-dimensional figure formed by three or more straight line segments. The word comes from Greek: "poly" means many and "gon" means angles.
Classification of Polygons by Number of Sides:
| Number of Sides | Name | Example |
|---|---|---|
| 3 | Triangle | Three-sided figure |
| 4 | Quadrilateral | Square, rectangle |
| 5 | Pentagon | Five-sided figure |
| 6 | Hexagon | Six-sided figure |
| 7 | Heptagon | Seven-sided figure |
| 8 | Octagon | Eight-sided figure |
| 9 | Nonagon | Nine-sided figure |
| 10 | Decagon | Ten-sided figure |
Types of Polygons based on Shape:
| Type | Description |
|---|---|
| Convex polygon | All interior angles < 180°, all diagonals lie inside |
| Concave polygon | At least one interior angle > 180°, at least one diagonal lies outside |
| Regular polygon | All sides equal AND all angles equal |
| Irregular polygon | Sides and/or angles are not all equal |
Key Properties of Polygons:
- A polygon has the same number of sides, vertices, and interior angles
- The sum of exterior angles of any polygon is always 360°
- A diagonal is a line segment joining two non-adjacent vertices
Example 1: What is each exterior angle of a regular hexagon?
360 ÷ 6 = 60°.
Example 2: What do the exterior angles of any polygon add up to?
360°.
Example 3: Name the polygon with: (a) 6 sides (b) 8 sides (c) 5 sides
- (a) 6 sides → Hexagon
- (b) 8 sides → Octagon
- (c) 5 sides → Pentagon
- **Answer:** (a) Hexagon, (b) Octagon, (c) Pentagon
*Example 2:
Classify the following as convex or concave:
A polygon where one interior angle is 210°.
Solution:
- If an interior angle is greater than 180°, the polygon is concave
- 210° > 180°, so the polygon is concave
- **Answer:** Concave polygon
*Example 3:
How many diagonals does a pentagon have?
Solution:
- Pentagon has 5 vertices
- Total line segments between vertices = \(\frac{5 \times 4}{2} = 10\)
- Subtract the 5 sides = \(10 - 5 = 5\) diagonals
- **Answer:** 5 diagonals
Quick recap
- Exterior angles of any polygon sum to 360°.
- Each exterior angle of a regular polygon = 360° ÷ n.
- A polygon is a closed figure with straight sides
- Polygons are named by number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), etc.
- Convex: all interior angles < 180°
- Concave: at least one interior angle > 180°
- Regular: all sides and all angles equal
- Number of diagonals = \(\frac{n(n-3)}{2}\) where \(n\) = number of sides
- ---
✓ Quick check
The exterior angles of any polygon add up to ___ ?
Exterior angles always sum to 360°.
Each exterior angle of a regular pentagon is ___ ?
360 ÷ 5 = 72°.
Sum of Interior Angles
Interior Angles of a Polygon: The angles inside a polygon formed by two adjacent sides are called interior angles.
Formula for Sum of Interior Angles: For a polygon with \(n\) sides:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
Each Interior Angle of a Regular Polygon:
\[ \text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n} \]
Exterior Angles of a Polygon: An exterior angle is formed by extending one side of a polygon. It is supplementary to the adjacent interior angle.
Important Properties:
- The sum of exterior angles of any polygon (taken one at each vertex) is 360°
- Each exterior angle of a regular polygon = \(\frac{360°}{n}\)
- Interior angle + Exterior angle = \(180°\) (they form a linear pair)
Relationship Table:
| Polygon (n sides) | Sum of Interior Angles | Each Interior Angle (Regular) | Each Exterior Angle (Regular) |
|---|---|---|---|
| Triangle (3) | 180° | 60° | 120° |
| Quadrilateral (4) | 360° | 90° | 90° |
| Pentagon (5) | 540° | 108° | 72° |
| Hexagon (6) | 720° | 120° | 60° |
| Octagon (8) | 1080° | 135° | 45° |
Example 1: Find the interior angle sum of a pentagon.
(5 − 2) × 180 = 540°.
Example 2: Find each interior angle of a regular hexagon.
(6 − 2) × 180 ÷ 6 = 720 ÷ 6 = 120°.
Example 3: Find the sum of interior angles of a hexagon.
- Hexagon has \(n = 6\) sides
- Sum = \((n - 2) \times 180° = (6 - 2) \times 180° = 4 \times 180° = 720°\)
- **Answer:** 720°
*Example 2:
Find the measure of each interior angle of a regular octagon.
Solution:
- Octagon has \(n = 8\) sides
- Sum of interior angles = \((8 - 2) \times 180° = 6 \times 180° = 1080°\)
- Each interior angle = \(\frac{1080°}{8} = 135°\)
- **Answer:** 135°
*Example 3:
The sum of interior angles of a polygon is 1440°. How many sides does it have?
Solution:
- \((n - 2) \times 180° = 1440°\)
- \(n - 2 = \frac{1440°}{180°} = 8\)
- \(n = 8 + 2 = 10\)
- **Answer:** 10 sides (decagon)
Quick recap
- Interior angle sum = (n − 2) × 180°.
- Each interior angle of a regular polygon = sum ÷ n.
- Sum of interior angles = \((n-2) \times 180°\)
- Each interior angle (regular) = \(\frac{(n-2) \times 180°}{n}\)
- Sum of exterior angles = \(360°\) (for any polygon)
- Each exterior angle (regular) = \(\frac{360°}{n}\)
- Interior and exterior angles are supplementary: \(I + E = 180°\)
- ---
✓ Quick check
What is the interior angle sum of a pentagon?
(5 − 2) × 180 = 540°.
What is each interior angle of a regular hexagon?
720 ÷ 6 = 120°.
Properties of Special Quadrilaterals
In a parallelogram, opposite sides and angles are equal and the diagonals bisect each other. A rectangle has equal diagonals; a rhombus has perpendicular diagonals; and a square has both.
Example 1: What is true of opposite angles in a parallelogram?
They are equal.
Example 2: What is special about the diagonals of a rhombus?
They are perpendicular.
Quick recap
- Parallelogram: opposite sides/angles equal, diagonals bisect.
- Rectangle: equal diagonals; rhombus: perpendicular diagonals; square: both.
✓ Quick check
The diagonals of a rectangle are ___ ?
A rectangle has equal diagonals.
A square's diagonals are equal and ___ ?
A square has equal and perpendicular diagonals.
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