Last modified on August 3rd, 2023

chapter outline



Once you have learned about points, line, line segments, and plane, the next thing to know is what happens when two lines meet at a point. That is when an angle is formed.

Angles are one of the basic concepts of geometry. We cannot think of defining any shape, be it triangles, quadrilaterals, or polygons, without their angles. An angle thus forms a part of each geometric shape.

What is an Angle

Mathematically, an angle is defined as a figure that forms when two rays meet at a common point. It is represented by the symbol ∠. An angle is usually measured in degrees, denoted with ‘ ° ‘. The term ‘angle’ comes from the Latin word ‘angulus’, meaning ‘corner’.


A degree is a measure of rotation.  A full rotation around a point gives us a full circle which measures 360°, a half rotation gives us a half-circle, which is 180°, and a quarter rotation gives us a right-angle measuring 90°.

Parts of an Angle

Parts of an Angle
  1. Arms: The two straight or curved sides that join to form the angle are called the arms. Here OX and OY are the arms of the angle ∠XOY.  
  2. Vertex: The common endpoint where the two rays meet to form an angle. Here the point ‘O’ is the vertex.

Naming an Angle

An angle can be named in two ways:

Method – 1: The angle symbol, followed by the three points that define the angle. The middle letter is the vertex. So the above figure can be written as ∠XOY or ∠YOX.

Method – 2: Just by writing the vertex. It is written as ∠O.

Sometimes, Greek letters such as α, β, γ, θ, and φ or lowercase Roman letters such as a, b, c, x, y, and z are also used to denote an angle or its size.

How to Find the Measure of an Angle

The size of an angle is measured using a protractor, which is usually semicircular in shape and is transparent.

Measuring Angles with a Protractor

Types of Angles

All angles are commonly classified based on their magnitude or degree of rotation, into six main types:

Types of Angles
  1. Acute Angle: An angle that measures less than 90° is called an acute angle. In other words, it lies between 0° to 90°.
  2. Right Angle: An angle that measures exactly 90° is called a right angle. It is formed when two sides of an angle are perpendicular to each other. The sign of a right angle can also be shown using a quarter circle along with the standard sign as shown in the figure.
  3. Obtuse Angle: An angle that measures greater than 90° and less than 180° is called an obtuse angle.
  4. Straight Angle: An angle that measures exactly 180° is called a straight angle. It is similar to a straight line and hence the name straight angle.
  5. Reflex Angle: An angle that measures greater than 180° and less than 360° is called a reflex angle.
  6. Complete Angle: An angle whose measure is equal to 360° is called a complete angle. It is formed by one complete rotation of one of its arms.

Other Types of Angles

Angles are also sometimes classified based on their position, the direction of rotation, the sum of their pairs, or their transversal into the following types:

  • Interior Angles
  • Exterior Angles
  • Positive Angles
  • Negative Angles
  • Complementary Angles
  • Supplementary Angles
  • Vertically Opposite Angles
  • Adjacent Angles
  • Alternate Interior Angles
  • Alternate Exterior Angles
  • Corresponding Angles
  • Consecutive interior Angles

Examples in Real Life

We see angles almost everywhere around us in our daily life. Some of the examples are given below:

  • Angles formed by the hands of a clock.
  • Alphabets A, K, M, N, V, W, X, Y, and Z contain acute angles while E, F, H, L, and T have the right angles.
  • Objects of daily use such as cloth hangers, scissors, the edge of tables, blades of a fan, cycle spokes, and wheels all have angles in them.
  • Study accessories such as a ruler, an opened compass, and set squares all contain angles.
  • Angles are used in the navigation of airplanes and ships.
  • Engineers and architects use angles to form the design of roads, buildings, and sports complexes.
  • Artists use their knowledge of angles to make sketches and paintings.

Last modified on August 3rd, 2023

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