Angles
Angles are formed when two lines intersect at a point. The measure of the 'opening' between these two rays is called an 'angle'. It is represented by the symbol ∠. Angles are usually measured in degrees and radians, which is a measure of circularity or rotation. Angles are a part of our day-to-day life. Engineers and architects use angles for the design of roads, buildings, and sporting facilities. Let us learn more about the definition of angles in Maths, the meaning of angles, the different properties of angles along with some angle examples.
| 1. | What are Angles? |
| 2. | Types of Angles and their Properties |
| 3. | Angles Based on Rotation |
| 4. | How to Measure an Angle? |
| 5. | How to Construct Angles? |
| 6. | FAQs on Angles |
What are Angles?
In geometry, an angle is formed when two rays are joined at their endpoints. These rays are called the sides or arms of the angle. Let us read about the different parts of an angle.
Parts of an Angle
There are two main parts related to an angle - the arms and the vertex.
Arms of the Angle
The two rays which join at a common point to form the angle are called the arms of the angle. Observe the figure given below which shows that OA and OB are the arms of the angle AOB.
Vertex of the angle
Vertex is a common endpoint that is shared by the two rays. Observe the figure in which the vertex O is marked as the joining point of the two arms.
Measure of an angle
An angle is measured in degrees. One full rotation around a point forms a complete angle of 360°.
The best way to measure an angle is by using a protractor. A protractor is a measuring instrument that is in the shape of a semi-circle. It is a translucent tool that helps us measure angles in degrees. It has degrees marked clockwise from 0° to 180° in the outer scale and anti-clockwise from 0° to 180° in the inner scale.
Types of Angles and their Properties
There are six types of angles. Each type of angle has a unique identification on the basis of angle measurement. Let us read about each type of angle individually along with their properties.
Acute Angle
An acute angle is an angle which is greater than 0° and less than 90°.
Right Angle
When an angle measures 90°, it is known as a right angle. A right angle can be easily observed as it forms the shape of the letter L.
Obtuse Angle
When an angle measures greater than 90° but less than 180°, it is an obtuse angle.
Straight Angle
The angle formed by a straight line is called a straight angle. In other words, a straight angle is a straight line, and the angle formed between two rays is equal to 180°. At a straight angle, the two rays are opposite to each other. Two right angles make up a straight angle. Since the measure of a straight angle is 180°, it is one-half of the whole turn of a circle.
Reflex Angle
A reflex angle is an angle whose measure is greater than 180° but less than 360°.
Complete Angle
When the measurement of an angle is equal to 360° it is a complete angle.
Angle Based on Rotation
Based on the direction of measurement or the direction of rotation, angles can be of two types:
- Positive Angles
- Negative Angles
Positive Angles
An angle measured in the counterclockwise (anti-clockwise) direction is a positive angle. In other words, positive angles are those angles that are rotated from the base in the anti-clockwise direction.
Negative Angles
Negative angles are those angles that are measured in a clockwise direction from the base. In other words, negative angles are those angles that are angles are rotated from the base in the clockwise direction.
How to Measure an Angle?
We use protractors to measure angles. Observe the figure given below which shows ∠AOB. Let us try and see if we can find out what type of angle is ∠AOB. Doesn't it look like an acute angle? This means that its measure is greater than 0° and less than 90°. Let us learn how to measure this angle using the protractor.
How to Measure an Acute Angle?
Let us try to measure the given ∠AOB.
- Step 1: Align the protractor with the ray OB as shown below. Start reading the inner scale from the 0° mark on the bottom-right of the protractor.
- Step 2: The number on the protractor that coincides with the second ray is the measure of the angle. Measure the angle using the inner scale of the protractor. Thus, ∠AOB = 37°
How to Measure an Obtuse Angle?
Now, let us try to measure the given ∠AOC.
- Step 1: Measure the angle using the outer scale of the protractor from the 0° mark on the bottom-left.
- Step 2: The number on the outer scale of the protractor that coincides with OA is the measure of ∠AOC. Thus, ∠AOC = 143°
How to Construct Angles?
We can construct angles using tools like a protractor or a compass. Here, let us use a protractor to construct angles. Let us draw an angle of 50°.
- Step 1: First, draw a ray OB and align the protractor with OB as shown.
- Step 2: Using the inner scale of the protractor, mark a point A above the marking on the protractor that corresponds to 50°.
- Step 3: Remove the protractor and draw a ray beginning at O that passes through this point A. Thus, ∠AOB is the required angle, that is, ∠AOB = 50°.
Note: If the ray extends in the other direction, we measure the angle using the outer scale from the 0° mark on the bottom-left.
The figure given below shows how to draw an angle of 50° when the ray is pointing in the other direction.
After placing the protractor on BO, we use the outer scale and mark 50° as shown. Then, we mark that point as A and join it to point O. This forms angle AOB = 50°
Important Notes on Angles
- 0°< Acute angle < 90°
- 90°< Obtuse angle < 180°
- 180° < Reflex angle < 360°
- A right angle is equal to 90°
- A straight angle is equal to 180°.
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Angles Examples
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Example 1: Observe the measure of the angles and identify the type of angles for each figure.
Solution:
a) The given angle is equal to 40°. It is an acute angle because its measure is less than 90°.
b) The given angle is equal to 117°. It is an obtuse angle because its measure is greater than 90° but less than 180°.
c) The given angle is equal to 121°. It is an obtuse angle because its measure is greater than 90° but less than 180°.
d) The given angle is equal to 185°. It is a reflex angle because its measure is greater than 180° but less than 360°.
FAQs on Angles
What is an Angle in Math?
Angles are formed when two rays intersect at a point. The 'opening' between these two rays is called an 'angle' which is represented by the symbol ∠. Angles are usually measured in degrees and are expressed as 60°, 90°, and so on.
What are the 6 Types of Angles?
The 6 types of angles are right angles, acute angles, obtuse angles, straight angles, reflex angles, and complete angles.
How do you Describe Angles?
An angle can be described as a figure formed by two rays meeting at a common endpoint called the vertex of the angle.
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What are the Types of Angles Based on Rotation?
Based on the direction of measurement or the direction of rotation, angles can be classified into two types:
- Positive Angles: Positive angles are those angles that are measured and rotated from the base in the anti-clockwise direction.
- Negative Angles: Negative angles are those angles that are measured and rotated from the base in the clockwise direction.
What is the Difference Between a Straight Angle and a Reflex Angle?
A straight angle is a straight line, and the angle formed between two rays is equal to 180°. It can be formed by combining two adjacent right angles. In other words, two right angles make up a straight angle. Whereas, a reflex angle is greater than 180° but less than 360°.
What are the Types of Angles Formed when a Transversal Passes through Parallel Lines?
When a transversal passes through parallel lines, many pairs of angles are formed, such as corresponding angles, vertically opposite angles, alternate interior angles, and alternate exterior angles.
What are the Types of Angles that Measure Less than 180°?
There are two types of angles that measure less than 180°, i.e., acute and obtuse angles. The measure of acute angles is always less than 90° while obtuse angles are more than 90° but always less than 180°. Examples of an acute angle are 50°, 60° and examples of obtuse angles are 170°, 165°.
What is the Sum of all the Three Angles of Triangle?
The sum of the three angles of a triangle is 180°.
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How Many 90 Degrees Angles are there in a Straight Angle?
There are two 90° angles in a 180-degree angle or a straight angle. Since 90° + 90° = 180°, therefore, two 90-degree angles are there in a straight angle.
List the Types of Angles that are Formed in Pairs.
The types of angles in pairs are listed below:
- Adjacent angles
- Complementary angles
- Supplementary angles
- Alternate interior angles
- Alternate exterior angles
- Corresponding angles
- Vertical angles
- Consecutive interior angles
What is a Complete Angle?
When an angle completes its full rotation starting from 0° and ends at 360° it is known as a complete angle. In other words, a complete angle measures 360°.
How do you Measure Angles?
Angles can be measured easily using a simple measuring instrument known as the protractor. This instrument has the shape of a semicircle which has an inner scale and an outer scale and with markings from 0° to 180° on it. A detailed explanation of measuring angles using a protractor is given above on this page.
How are Angles Used in Real Life?
There are many places where angles are used. In real life, angles are commonly used in the construction of buildings, They act as a crucial factor in the stability and construction of buildings because inaccuracy in these fields might lead to damages and unknown and unfortunate incidents like the collapse of buildings, the gaps can give way to leakages and many other instabilities.