Basic Terms Related to Angle: The most common word used in more than one chapter of Mathematics is ‘angle’. We have been using it verbally, in construction, and in calculations for many years now. Angles are widely seen in clocks, corners of rooms, and edges of books. They are the prime requirement for constructing anything.

For example, engineers and architects use angles the most. They use tools like compass and protractor to create and measure angles in their designs. Angles are used to design and construct buildings, houses, bridges, and smaller objects like furniture.

What is an Angle?

Angle is the measure of rotation of a ray about its origin. Angles are measured in degrees or radians.

What are the Parts of an Angle?

• The ray in its starting position is called the initial side.
• After being rotated about the initial point, the ray in its new position is called the terminal side.
• The point of rotation of the ray is called the vertex.

Here, \(OY\) is the initial side of the angle, and \(OX\) is the terminal side. While the initial side is stationary, the terminal side is rotated about \(O\) to form the required angle.

How are Angles Denoted?

There are three common ways to represent an angle.

Method 1: Use the \(^ \wedge \) (cap symbol) with the name of the vertex.

Method 2: Use the \(\angle \) (angle symbol) along with the name of the vertex.

Method 3: Use the \(\angle \) (angle symbol) along with the three names that define the conjoining rays. The vertex where the angle is formed is the centre letter in this representation.

For example, observe this angle.

This angle can be referred to as \(\hat B\), or \(\angle B\), or \(\angle ABC\), or \(\angle CBA\).

Positive and Negative Angles

As we know that a ray is rotated about the vertex to form various measures of angles, the direction of rotation influences the sign of the angle measure.

• When a ray is rotated in the clockwise direction, the angle measured is negative.
• The angle measured is positive when a ray is rotated in the counter-clockwise direction.

How are Angles Measured?

An angle is the measure of the opening between the two rays. The most common tool to measure an angle is the protractor. A protractor measures the angles in degrees.

Degree Measure

We know that a whole angle is achieved by moving the initial side for one complete revolution. When the rotation of the initial side to the terminal side is \({\frac{1}{{360}}^{th}}\) of a full revolution, the angle made is said to measure \(1\) degree. The symbol for the degree is \(^{\rm{o}}\).

Each degree is further split into \(60\) minutes and each minute into \(60\) seconds. This can be written as,

\({1^{\rm{o}}} = {60^\prime }\)

\({1^\prime } = {60^{\prime \prime }}\)

Radian Measure

Angles are also measured in another unit called radian. One radian is the angle measure subtended by an arc of unit length at the centre of a unit circle. Radians are also the \({\rm{SI}}\) unit of measurement of angles.

Here, \(OA = OB\) is the radius of the circle that is \(1\) unit. \(AB\) is the arc of unit length. Then, \(\angle O = 1\) radian.

We know that perimeter of a circle \( = 2\pi r\)

Hence, the length of the arc, \(l = r \times \theta \)

Where,

\(r \to \) radius of the circle

\(\theta \to \) angle subtended at the centre in radians, \(\theta < 2\pi \)

Relation between Degree and Radian

We know that one complete revolution covers an angle of \({360^{\rm{o}}}\) or \(2\pi \) radians.

\(\therefore \,{360^{\rm{o}}} = 2\pi \)

\( \Rightarrow \pi = {180^{\rm{o}}}\)

Mathematically, \(\pi = \frac{{22}}{7}\) then, we have,

\(1\,{\rm{radian}} = \frac{{{{180}^{\rm{o}}}}}{\pi } = \frac{{{{180}^{\rm{o}}}}}{{\frac{{22}}{7}}} = {57^{\rm{o}}}{16^\prime }\)

Similarly,

\({1^{\rm{o}}} = \frac{\pi }{{{{180}^{\rm{o}}}}} = \frac{{\frac{{22}}{7}}}{{180}} = 0.01746\) radian

Given below are the degree and angle measures of some common angles.

Degree	Radian
\({{{30}^{\rm{o}}}}\)	\(\frac{\pi }{6}\)
\({{{45}^{\rm{o}}}}\)	\(\frac{\pi }{4}\)
\({{{60}^{\rm{o}}}}\)	\(\frac{\pi }{3}\)
\({{{90}^{\rm{o}}}}\)	\(\frac{\pi }{2}\)
\({{{180}^{\rm{o}}}}\)	\({\pi }\)
\({{{270}^{\rm{o}}}}\)	\(\frac{{3\pi }}{2}\)
\({{{360}^{\rm{o}}}}\)	\({2\pi }\)

Note: While writing in radian measure, the term ‘radian’ is often not included.

Conversion of Units

Radian measure \( = \frac{\pi }{{180}} \times \) Degree measure

Degree measure \( = \frac{{180}}{\pi } \times \) Radian measure

Steps to Convert Degrees to Radians

Step 1: Write the given angle in degrees.

Step 2: Multiply by \(\frac{\pi }{{180}}\). The operation is similar to the multiplication of fractions.

Step 3: Simplify.

What are the Types of Angles?

There are \(6\) types of angles based on the rotation of the arms of the angle.

1. Acute Angle: when the terminal side is rotated to a measure less than \({{{90}^{\rm{o}}}}\).
2. Right Angle: when the terminal side is rotated to a measure that is exactly \({{{90}^{\rm{o}}}}\).
3. Obtuse Angle: when the measure of rotation of the terminal side is more than \({{{90}^{\rm{o}}}}\).
4. Straight Angle: when measure of rotation is exactly \({{{180}^{\rm{o}}}}\).
5. Reflex Angle: when the measure of rotation is more than \({{{180}^{\rm{o}}}}\) but less than \({{{360}^{\rm{o}}}}\).
6. Complete Angle: when the measure of rotation is exactly \({{{360}^{\rm{o}}}}\).

Angle Pairs

When two angles occur together and exhibit some geometrical property, they are called angle pairs. Some common angle pairs are described below.

Complementary Angles

Two angles that add up to make a right angle are known as complementary angles.

Example: \({{{50}^{\rm{o}}}}\) and \({{{40}^{\rm{o}}}}\)

Supplementary Angles

Two angles that make a sum of \({{{180}^{\rm{o}}}}\) are said to be supplementary to each other.

Example: \({{{115}^{\rm{o}}}}\) and \({{{65}^{\rm{o}}}}\)

Vertical Opposite Angles

When two lines intersect, the angles that are opposite to each other are called vertically opposite angles. This angle pair is always congruent, i.e. two vertically opposite angles have the same measure.

Example:

Here, vertically opposite angles are:

• \(\angle m\) and \(\angle n\)
• \(\angle a\) and \(\angle b\)

This also means that:

• \(\angle m = \angle n\)
• \(\angle a = \angle b\)

Alternate Interior Angles

This angle pair is formed when a transversal intersects two parallel lines. The angles that lie between the parallel lines and on the opposite sides of the transversal. Alternate angles are congruent.

Example:

Here, alternate interior angles are:

• \(\angle 3\) and \(\angle 6\)
• \(\angle 5\) and \(\angle 4\)

Alternate Exterior Angles

Angles that lie outside the parallel lines and on the opposite sides of the transversal are called alternate exterior angles. This angle pair is congruent.

Here, alternate exterior angles are:

• \(\angle 2\) and \(\angle 7\)
• \(\angle 1\) and \(\angle 8\)

Corresponding Angles

When a transversal intersects two parallel lines, the angles that lie in the same relative position at each intersection are called corresponding angles.

Here, corresponding angles are:

• \(\angle 1\) and \(\angle 6\)
• \(\angle 2\) and \(\angle 5\)
• \(\angle 3\) and \(\angle 8\)
• \(\angle 4\) and \(\angle 7\)

Note that corresponding angles are congruent.

Adjacent Angles

Two angles with the same vertex and share a common side are called adjacent angles.

Solved Examples – Basic Terms Related to Angle

Q.1. Find the complementary and supplementary angle of \({65^{\rm{o}}}\).
Ans:
Given: \(\angle A = {65^{\rm{o}}}\)
Sum of complementary angles \( = {90^{\rm{o}}}\)
\(\therefore \) The complementary angle of \({65^{\rm{o}}} = {90^{\rm{o}}} – {65^{\rm{o}}} = {25^{\rm{o}}}\)
Similarly,
Sum of supplementary angles \( = {180^{\rm{o}}}\)
\(\therefore \) The supplementary angle of \({65^{\rm{o}}} = {180^{\rm{o}}} – {65^{\rm{o}}} = {115^{\rm{o}}}\)

Q.2. Find the unknown angles in the diagram.

Ans:
Given: \(\angle 7 = {45^{\rm{o}}}\)
\(\angle 4 = 5x + {35^{\rm{o}}}\)
Since vertically opposite angles are equal,
\(\angle 7 = \angle 6 = {45^{\rm{o}}}\)
Since alternate interior angles are equal,
\(\angle 6 = \angle 3 = {45^{\rm{o}}}\)
\( \Rightarrow \angle 3 = \angle 2 = {45^{\rm{o}}}\) (Vertically opposite angles)
\(\angle {45^{\rm{o}}}\) and \(\angle 8\) lie on a straight line. They are supplementary.
\(\therefore \,\angle 8 = {180^{\rm{o}}} – {45^{\rm{o}}}\)
\( \Rightarrow \angle 8 = {135^{\rm{o}}}\)
Therefore, by the same means, we can say that,
\(\angle 8 = \angle 5 = \angle 1 = \angle 4 = {135^{\rm{o}}}\)

Q.3. Convert \({120^{\rm{o}}}\) to radians.
Ans:
Step 1: Write the given angle in degrees.
Given: \({120^{\rm{o}}}\)
Step 2: Multiply by \(\frac{\pi }{{180}}\).
\(120 \times \frac{\pi }{{180}} = \frac{{120\pi }}{{180}}\)
Step 3: Simplify.
\(\frac{{120\pi }}{{180}} = \frac{{2\pi }}{3}\)

Q.4. Convert \(\frac{{5\pi }}{2}\) into degrees.
Ans:
Degree measure \( = \frac{{180}}{\pi } \times \frac{{5\pi }}{2}\)
\( = 180 \times \frac{5}{2}\)
\( = {450^{\rm{o}}}\)

Q.5. Find the length of the arc on a circle of radius \(5\,{\rm{cm}}\), subtending an angle of \({15^{\rm{o}}}\) at the centre.
Ans:
Arc length, \(l = r\theta \)
Here,
\(r = 5\;\,{\rm{cm}}\)
\(\theta = {15^{\rm{o}}} = \frac{\pi }{{12}}\)
\(\therefore \,l = 5 \times \frac{\pi }{{12}}\)
\(l = \frac{{5\pi }}{{12}}\)

Summary

Angles are geometric figures formed by rotating a ray about its vertex. They are measured using a tool called a protractor. The SI unit of measurement of angles in radians. The other standard unit of angle measure is degrees. These units of angles are interrelated and convertible. There are different angle types based on the rotation. When angles occur in twos, they are called angle pairs. There are different types of angle pairs based on their position. The angles that share a vertex and have a common side are complementary and supplementary.

Frequently Asked Questions (FAQs)

Q.1. How are angles formed?
Ans: Angles are formed when a ray from its initial position is rotated about its origin. It is the measure of rotation of the ray to its terminal position.

Q.2. What are the types of angles based on rotation?
Ans: The types of angles based on rotation are:

• Acute angle
• Right angle
• Obtuse angle
• Straight angle
• Reflex angle
• Full angle

Q.3. What is the basic unit of an angle?
Ans: Although the \({\rm{SI}}\) unit of angle measure is radians, the most common unit is degrees.

Q.4. How are angles named?
Ans: There are three ways to name an angle:

• Use the \(^ \wedge \) (cap symbol) with the name of the vertex.
• Use the \(\angle \) (angle symbol) along with the name of the vertex.
• Use the \(\angle \) (angle symbol) along with the three names that define the conjoining rays.

Q.5. What are radians and degrees?
Ans: These are the two units to measure angles. One complete revolution of ray from its original position makes \({360^{\rm{o}}}\) which is equivalent to \(2\pi \). Hence, we can say that
\(2\pi = {360^{\rm{o}}}\)
\( \Rightarrow \pi = {180^{\rm{o}}}\)
\(\therefore \,1\,{\rm{radian}} = \frac{{{{180}^{\rm{o}}}}}{\pi }\)

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