Angle Definition
Have you seen a kite? Apart from the vibrant colours in it, have you noticed that each side of it makes an angle with another part at the corner? From the image below, you can see that whenever two lines are intersecting, an angle is formed between the lines.
Learning about angles is essential as it forms the foundation of geometry and other mathematical concepts.
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You can find angles in many other things that you use in your everyday life like a pair of scissors, a chair, a hockey stick, etc.
In this article, we will take a close look at the definition of angle in maths, various concept of angles, and the different types of angle that exist in mathematics.
What is an Angle
The word angle finds its origin from a Latin word “Angulus” whose meaning is “a small bending” and the concept of angles was introduced by Eudemus (an ancient Greek philosopher) who gave the definition of an angle as a deviation from a straight line.
So, what is the definition of angle in mathematical terms? When two lines meet at a single shared point, they form an angle. The lines forming an angle don’t need to be only straight lines, an angle can also be formed by the intersection of curved lines. Another angle definition could be the degree of turn of a line around a central point.
To clarify the angle meaning better, let us look at various parts of an angle with the help of a diagram:
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Arm - The two rays that come together to form the angle are called arms of the angle or sides of the angle. So in the above diagram, OB and OA are the arms of the angle AOB.
Vertex - The endpoint, which is common to both the arms of an angle, is called the vertex of the angle. In the given figure, O is the vertex of angle AOB.
Angles are measured in degrees and represented by the symbol °.
Types of Angles
There are various types of angles that are based on the measure of their degrees. With the increase in the angle, its name changes.
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Acute Angle
If you look at the letter “A” in the word “Acute”, the angle between the lines is an acute angle. Any angle which is less than 90 degrees is an acute angle. Their value lies between 0 degrees and 90 degrees i.e. 0° < 𝛉 < 90°.
Right Angle
An angle that is equal to 90 degrees is a right angle. It is mostly formed by two perpendicular lines and 𝛉 = 90°.
Obtuse Angle
An angle that is greater than 90 degrees and less than 180 degrees is an obtuse angle. Its value lies between 90 degrees and 180 degrees i.e. 90° < 𝛉 < 180°.
In a straight line, Obtuse angle = 180 - acute angle.
Straight Angle
A straight angle is exactly 180 degrees. Since it is similar to a straight line, that is why it is termed as the straight angle. A straight angle is a mixture of acute and obtuse angles on a straight line, i.e. Straight angle = Obtuse angle + acute angle.
Reflex Angle
An angle that is greater than 180 degrees and less than 360 degrees is an obtuse angle. Its value lies between 180 degrees and 360 degrees i.e. 180° < 𝛉 < 360°. You can calculate the reflex angle if you know the value of the acute angle since it is the complementary angle to the acute angle on the other side of the straight line.
Complete Angle
If you rotate a straight line so that it finally reaches its original starting position, it forms a complete angle that is equal to 360 degrees. This is referred to as the complete angle or a full angle. Hence 𝛉 = 360 degrees here. Another name for this angle is perigon, and it is the central angle of a circle.
Complete angle = four right angles = two straight angles.
Relationships Between Angles
Angles are related to each other based on either their summation or where they lie on two intersecting lines:
Complementary Angles
Two angles are called complementary angles to each other if they sum up to 90 degrees. It is not necessary for the angles to be adjacent to each other to be called complementary angles. As long as their summation is 90 degrees, they are termed complementary angles. In the diagram below, both sets of angles are complementary. In the first one, they are adjacent to each other while in the second one they are not.
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Supplementary Angles
Supplementary angles are angles whose sum is 180 degrees. Supplementary angles could be of different types as described below:
Vertical angles - When two lines cross, the angles opposite to each other are called vertical angles. Vertical angles are equal in measure. In the figure below, angles 1 and 3 are vertical angles and equal to each other. Angles 2 and 4 also form vertical angles and are equal to each other.
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Alternate interior angles - When we draw a traversal to two straight lines, the angles on the opposite sides of the traversal that lie on the interior side constitute the alternate interior angles. If the two lines are parallel to each other, then alternate interior lines are equal. Angles 2 and 3 are alternate interior angles in the image below.
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Alternate exterior angles - When we draw a traversal to two straight lines, the angles on the opposite sides of the traversal that lie on the exterior side constitute the alternate exterior angles. If the two lines are parallel to each other, then alternate exterior lines are equal. Angles 1 and 4 are alternate exterior angles here.
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Corresponding angles - When a line crosses two lines, the angles in matching corners are called corresponding angles. One of them is internal, and another is external. Corresponding angles are equal if the two lines (which are intersected) are parallel to each other. In the image below angles, 1 and 2 are corresponding angles. Here 1 is the external angle, and 2 is the internal angle.
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Adjacent angles - Two angles with a common vertex and a side are called adjacent angles.
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FAQs on Angles - Definition and It's Types
1. What is an angle in geometry?
In geometry, an angle is formed when two rays originate from a common endpoint. The rays are called the arms or sides of the angle, and the common endpoint is known as the vertex. Angles measure the amount of turn between the two arms and are typically measured in degrees (°).
2. What are the main types of angles based on their measurement?
Based on their degree measurement, angles are primarily classified into six types:
- Acute Angle: An angle that measures less than 90°.
- Right Angle: An angle that measures exactly 90°.
- Obtuse Angle: An angle that measures more than 90° but less than 180°.
- Straight Angle: An angle that measures exactly 180°, forming a straight line.
- Reflex Angle: An angle that measures more than 180° but less than 360°.
- Complete Angle: An angle that measures exactly 360°, representing a full rotation.
3. Where can we see examples of different angles in everyday life?
Angles are all around us. For example:
- The blades of an open pair of scissors form an acute angle.
- The corner of a book or a square tile forms a right angle (90°).
- A laptop screen tilted back or a reclining chair forms an obtuse angle.
- The hands of a clock at 6:00 form a straight angle (180°).
4. What is the difference between complementary and supplementary angles?
The key difference lies in their sum:
- Complementary angles are two angles whose measures add up to 90°. They do not have to be adjacent. For example, a 30° angle and a 60° angle are complementary.
- Supplementary angles are two angles whose measures add up to 180°. A pair of angles that form a straight line are always supplementary. For example, a 110° angle and a 70° angle are supplementary.
5. How are angles related when a transversal line intersects two parallel lines?
When a transversal line cuts across two parallel lines, several important angle relationships are formed:
- Corresponding angles are in the same position at each intersection and are equal.
- Alternate interior angles are on opposite sides of the transversal, between the parallel lines, and are equal.
- Alternate exterior angles are on opposite sides of the transversal, outside the parallel lines, and are equal.
- Vertically opposite angles at each intersection are equal.
6. Can an angle be larger than a straight angle? Explain with an example.
Yes, an angle can be larger than a straight angle (180°). This type of angle is called a reflex angle. It measures more than 180° but less than 360°. For instance, if you measure the larger angle around the outside of the hands of a clock at 1:00, it would be a reflex angle of 330°.
7. What are vertically opposite angles and what is their key property?
Vertically opposite angles are the angles that are opposite each other when two lines intersect, forming an 'X' shape. The most important property of vertically opposite angles is that they are always equal to each other. For example, in an 'X', the top and bottom angles are equal, and the left and right angles are equal.
8. Why is a full circle or a complete turn considered 360 degrees?
A full circle is considered 360° because it represents a complete rotation. If a ray rotates around its endpoint and returns to its starting position, it has traced a full circle. This complete turn is divided into 360 equal parts, with each part being one degree. This convention allows for the straightforward definition of other angles, such as a right angle being a quarter turn (90°) and a straight angle being a half turn (180°).
