What is an Angle?
Two rays or lines meeting at a common point lead to the formation of an angle. An angle is represented using the symbol, ‘’.
Angles are measured in two different units and they are:
- Degrees (°)
- Radians (in terms of )
We use a protractor to measure angles and this instrument helps in measuring angles in degrees (°).
What are the Parts of an Angle?
An angle consists of a vertex and two sides. The sides and vertex of an angle can be described as:
- The vertex of an angle is the point at which two lines or rays meet.
- Angle sides are the two rays or lines that meet to form the angle.
The angle formed by the rays AB and BC at the vertex B as shown in the image can be written as ‘∠ABC’, or ∠CBA, or simply as ∠B.
AB and AC are the sides of ∠ABC, and B is the vertex of ∠ABC.
How do we Measure an Angle?
The following steps can be used to measure angles with a protractor:
Step 1: Place the center of the protractor at the vertex of an angle.
Step 2: Align the baseline of the protractor with one side of the angle.
Step 3: Another side of the angle on the protractor indicates the measure of the angle in degrees. Measure anticlockwise if you have chosen the right baseline, or clockwise if you have chosen the left baseline of the protractor.
AMC = 40°
Types of Angles
There are four types of angles:
- Right angle
- Straight angle
- Acute angle
- Obtuse angle
Let’s study about each of these types in more detail.
Right angle
A right angle is an angle that measures 90°. It resembles the letter L in the English alphabet.
Straight angle
A straight angle is an angle that measures 180°. It looks like a straight line.
Acute Angle
An acute angle is an angle that measures less than a right angle. In mathematical terms, an angle measuring less than 90° is called an acute angle.
Examples of acute angles include 85°, 5°, 71°, 22° and so on.
An acute angle of 45° is shown in the image.
Obtuse Angle
An obtuse angle is an angle that measures greater than a right angle but less than a straight angle. In mathematical terms, an angle measuring between 90° and 180° is called an obtuse angle.
Examples of obtuse angles include 166°, 121°, 117° and so on.
Solved Examples
Example 1: Identify the type of angle shown in the image.
Solution:
The angle appears to be less than a right angle.
Hence, the angle is an acute angle.
Example 2: What is the name of the angle? Classify the angle.
Solution:
The angle is L-shaped, so it is a right angle.
The vertex at which the angle is formed is X, hence the angle can be written as:
∠X = 90°.
Example 3: How many right angles are in a straight angle?
Solution:
Measure of straight angle = 180°
= 90° + 90° [Write 180 as sum of 90 and 90]
We know that 1 right angle measures 90° and 180° can be split into two 90° angles, as shown above as the sum of two right angles.
Hence, a straight angle has two right angles.
Example 4: What is the measure of angle AOC?
Solution:
∠AOB=∠BOC + ∠AOC [Write equation to find unknown angle]
90°=30° + ∠AOC [Substitute values]
90°-30°=30° + ∠AOC-30° [Subtraction property of equality]
60°= ∠AOC [Simplify]
∠AOC=60° [Rewrite equation]
Hence, ∠AOC measures 60°.
A protractor is a commonly used instrument in geometry and its purpose is to measure angles. It is semicircular in shape, and is usually made of plastic or glass that makes it transparent.
A right angle measure is exactly 90 degrees.
The parts of an angle include rays or sides and a common vertex.
Radian is a measuring unit for angles used in many mathematical fields. It is denoted by rad.