Angle Properties – O Level Exam Preparation Guide

In this revision note, you will learn the all the angle properties and applying them to solve Geometry questions on Polygons.

Different types of angles

a) An acute angle is less than 90°.

acute angle

b) A right angle is equal to 90°.

right angle

c) An obtuse angle is between 90° and 180°.

obtuse angle

d) A straight line is equal to 180°.

angles on a straight line

e) A reflex angle is larger than 180° but less than 360°.

reflex angle

f) A complete turn is 360°

angles at a point

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Angle Properties

1) Two angles are said to be complementary angles if their sum is 90°.

2) Two angles are called supplementary angles if their sum is 180°.

3) Vertically opposite angles are those on the opposite sides of two intersecting lines.

vertically opposite angles

4) Two angles are said to be corresponding angles if they have the same relative positive at the each intersection where a line cuts across a pair of parallel lines.

corresponding angles

The corresponding angles are equal to each other.

5) Two angles are called alternate angles if they lie on the different side of the line cutting across a pair of parallel lines.

alternate angles

The alternate angles are equal to each other.

 

6) Two angles are known as interior angles if they lie on the same side of the line that lie on the different side of the line cutting across a pair of parallel lines.

interior angles

Interior angles add up to 180°.

Angle Properties of Polygons

A polygon is a closed plane figure with three or more straight lines. Polygons are named according to the number of sides they have.

names of polygons

Note: Polygon with n sides is called an n-gon.

A regular polygon is a polygon with all its sides and angles equal.

 

Angle Properties of Triangles

a) An equilateral triangle is a triangle with 3 equal sides. Each of its angles is 60°.

equilateral triangle

b) An isosceles triangle is a triangle with 2 equal sides. Its base angles are equal.

isosceles triangle

The sum of the interior angles of a triangle is always 180°.

Angle Properties of Rhombus

angle properties of rhombus

  • Opposite angles are equal ( ∠a  = ∠a )
  • ∠a + ∠b = 180° (Interior angles)

rhombus

  • The diagonals bisect the angles equally
  • The diagonals intersect at Right Angles

Angle Properties of Parallelogram

angle properties of parallelogram

  • Opposite angles are equal (∠a = ∠a )
  • ∠a + ∠b = 180° (Interior angles) 

paralleogram

  • diagonals DO NOT bisect the angles equally
  • diagonals DO NOT intersect at Right Angles

Angle Properties of Trapezium

angle properties of trapezium

  • ∠a + ∠d = 180° (interior angles formed between parallel lines)
  • ∠b + ∠c = 180° (interior angles formed between parallel lines)
  • ∠a + ∠d = ∠b + ∠c

Sum of Angles in a Quadrilateral

A quadrilateral is a 4-sided polygon. Any quadrilateral can be divided into 2 triangles along its diagonal. Since the angles of a triangle add up to 180°, the angle sum of a quadrilateral is 180° × 2 = 360°.

Sum of Interior Angles in a Polygon

A polygon with n sides has n interior angles.

The sum of interior angles = (n – 2) × 180°.

Sum of Exterior Angles in a Polygon

A polygon with n sides has n exterior angles.

Each pair of interior angle and exterior angle is supplementary (add up to 180°).

The sum of the exterior angles of an n-sided polygon is always 360°

Angle Properties of Polygons Example 1

Find the exterior angle of a 12-sided regular polygon.

Let x the size of each exterior angle.

12 × x = 360°

x° = 30°

Angle Properties of Polygons Example 2

The ratio of an interior angle to an exterior angle of a regular n-sided polygon is 8 ∶ 1. How many sides does the polygon have?

Let 8a and a be the interior angle and the exterior angle at the same vertex respectively.

exterior angle + interior angle = 180° (adjacent ∠s on a straight line)

a + 8a = 180°

a = 20°

Sum of exterior angles = 360°

20°(n) = 360°

n = 18

∴ The polygon has 18 sides.

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