9.1 Polygons
A polygon is a closed 2D plane shape bounded by three or more straight side line segments.
Names and Number of Sides of Polygons
- Triangle: 3 sides, 3 vertices, 0 diagonals
- Quadrilateral: 4 sides, 4 vertices, 2 diagonals
- Pentagon: 5 sides, 5 vertices, 5 diagonals
- Hexagon: 6 sides, 6 vertices, 9 diagonals
- Heptagon: 7 sides, 7 vertices, 14 diagonals
- Octagon: 8 sides, 8 vertices, 20 diagonals
- Nonagon: 9 sides, 9 vertices, 27 diagonals
- Decagon: 10 sides, 10 vertices, 35 diagonals
Formula for Number of Diagonals
For an $n$-sided polygon, the number of diagonals is given by:
$$\text{Number of diagonals} = \frac{n(n - 3)}{2}$$
9.2 Properties of Triangles and Angles
Classification of Triangles by Sides
- Equilateral Triangle: All 3 sides are equal; all 3 interior angles are $60^\circ$; 3 lines of symmetry.
- Isosceles Triangle: 2 sides are equal; 2 base angles are equal; 1 line of symmetry.
- Scalene Triangle: All 3 sides and angles are different; 0 lines of symmetry.
Classification of Triangles by Angles
- Acute-angled Triangle: All three angles are acute ($< 90^\circ$).
- Right-angled Triangle: One interior angle is exactly $90^\circ$.
- Obtuse-angled Triangle: One interior angle is obtuse ($> 90^\circ$).
Angle Properties of Triangles
- Sum of Interior Angles: $a + b + c = 180^\circ$
- Exterior Angle Property: An exterior angle is equal to the sum of its two opposite interior angles ($x = a + b$).
- Sum of Interior and Adjacent Exterior Angle: $180^\circ$
9.3 Properties of Quadrilaterals and Angles
Types of Quadrilaterals and Properties
- Square: All 4 sides equal, all angles $90^\circ$, diagonals bisect each other at $90^\circ$, 4 lines of symmetry.
- Rectangle: Opposite sides equal and parallel, all angles $90^\circ$, diagonals equal in length, 2 lines of symmetry.
- Parallelogram: Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other, 0 lines of symmetry.
- Rhombus: All 4 sides equal, opposite sides parallel, opposite angles equal, diagonals bisect each other at $90^\circ$, 2 lines of symmetry.
- Trapezium: Only one pair of opposite sides is parallel, 0 or 1 line of symmetry.
- Kite: Two pairs of adjacent sides equal, one pair of opposite angles equal, diagonals intersect at $90^\circ$, 1 line of symmetry.
Angle Properties of Quadrilaterals
- Sum of Interior Angles: The sum of interior angles of any quadrilateral is $360^\circ$.
- Interior Angles on the Same Side of Parallel Sides: Add up to $180^\circ$.