Pair of Angles: Definition, Diagrams, Types, and Examples

An angle is formed when the two rays are combined at a common point. The common point here is the vertex, and the two rays are known as the arms of the angle. The symbol \(‘\angle ‘\) represents the angle. In geometry, we often come across pairs of angles that have been given specific names. In this section, we shall learn about such pair of angles. 

What is an Angle? 

Angles are formed when lines or line segments meet. And the corners or vertex are formed when two lines or line segments intersect at a point. In daily life, you get to see various angles formed between the edges of the plane surfaces. To make a similar model using the plane surface, you need to understand angles thoroughly.

Definition

An angle is formed when the two rays originate from the same originating point. The rays making an angle are known as the arms of the angle, and the originating point is known as the vertex of the angle.

The symbol represents the angle \(\angle \). Here in the diagram, the angle formed is represented as \(\angle PQR\)

Learn Construction of Angles here

Pair of Angles

In geometry, we often come across pairs of angles that have been given specific names. In this, we will discuss pair of angles:

Adjacent angles: Two angles in a plane are known as the adjacent angles if

a) they have a common vertex.

b) they have a common arm and 

c) their other arms lie on the opposite sides of the common arm.

In the given diagram, \(\angle AOC\) and \(\angle BOC\) have the common vertex \(O.\) Also, they have the common arm \(OC\) and their other arms \(OA\) and \(OB\) lie on the opposite sides of the common arm \(OC.\)

Therefore, \(\angle AOC\) and \(\angle BOC\) are the adjacent angles.

Note: That \(\angle AOC\) and \(\angle AOB\) are not adjacent angles because their other arms \(OB\) and \(OC\) are not on the opposite side of the common arm \(OA.\)

Linear Pair

Two adjacent angles are said to form a linear pair of angles if their non-common arms are two opposite rays.

In the given diagram, \(OA\) and \(OB\) are two opposite rays and \(\angle AOC\) and \(\angle BOC\) are the adjacent angles. Therefore, \(\angle AOC\) and \(\angle BOC\) form a linear pair.

If you measure \(\angle AOC\) and \(\angle BOC\) with the help of the protractor, you will find the sum of their measure equal to \({180^ \circ }\)

Thus, the sum of the angles in a linear pair is \({180^ \circ }\)

Vertically Opposite Angles

Two angles formed by two intersecting lines having no common arm are known as vertically opposite angles.

In the given diagram, two lines \(AB\) and \(CD\) are intersecting at a point \(O.\) We observe that with the intersection of these lines, four angles have been formed. Angles \(\angle 1\) and \(\angle 3\) form a pair of vertically opposite angles, while angles \(\angle 2\) and \(\angle 4\) form another pair of vertically opposite angles.

Angles \(\angle 1\) and \(\angle 2\) form a linear pair.

\(\angle 1 + \angle 2 = {180^ \circ } \Rightarrow \angle 1 = {180^ \circ } – \angle 2\) (1)

Also, \(\angle 2\) and \(\angle 3\) form a linear pair.

\(\therefore \angle 2 + \angle 3 = {180^ \circ } \Rightarrow \angle 3 = {180^ \circ } – \angle 2\) (2)

From \((1)\) and \((2),\) we get \(\angle 1 = \angle 3\)

Similarly, we can prove that \(\angle 2 = \angle 4\)

Thus, if two lines intersect, then vertically opposite angles are always equal.

In the given diagrams below, \(\angle 1\) and \(\angle 2\) are not vertically opposite angles because their arms do not form two pairs of opposite rays.

Pair of Supplementary Angles

Supplementary Angles

The two angles whose sum is \({180^ \circ }\) are called supplementary angles. When the two angles are supplementary, each angle is known to be the supplement of the other. As shown in the diagram below, \({60^ \circ }\) angle is the supplement of \({120^ \circ }\) angle or vice versa because their sum is \({180^ \circ }\)

Angles of \({55^ \circ }\) and \({125^ \circ }\) are supplementary angles.

The supplementary of an angle of \({130^ \circ }\) is the angle of \({50^ \circ }\) and, the supplement of an angle of \({50^ \circ }\) is the angle of \({130^ \circ }\)

Observations:

Two acute angles cannot be the supplement of each other.

Two right angles are always supplementary.

Two obtuse angles cannot be supplementary to each other.

Note that the angles of a linear pair are always supplementary. But supplementary angles need not always form a linear pair. The difference is that the linear pair of angles must have their vertices common. But for two angles to be supplementary angles, they may or may not share a common vertex. The vertices of the two supplementary angles may be different.

Pair of Complementary Angles

Complementary Angles

The two angles whose sum is \({90^ \circ }\) are called complementary angles. Whenever the two angles are known to be complementary, each angle is said to complement the other. As shown in the diagram below, \({30^ \circ }\) angle is the complement of \({60^ \circ }\) angle or vice versa because their sum is \({90^ \circ }\)

Angles of measures \({35^ \circ }\) and \({55^ \circ }\) are complementary angles. The angle of \({35^ \circ }\) is the complement of the angle of \({55^ \circ }\) and the angle of \({55^ \circ }\) is the complement of the angle of \({35^ \circ }\)

The complement of an angle of measure \({30^ \circ }\) is the angle of \({60^ \circ }\). And, the complement of the angle of measure \({60^ \circ }\) is the angle of \({30^ \circ }\)

Observations:

If two angles are the complement of each other, then each is an acute angle. But any two acute angles need not be complementary. For example, angles of measure \({30^ \circ }\) and \({50^ \circ }\) are not complemented by each other.

Two obtuse angles cannot be a complement to each other.

Two right angles cannot be the complement each other

Solved Examples – Pair of Angles

Q.1. Let’s measure the complement of each of the given angle: 60° 
Ans:  The given angle is \({60^ \circ }\)
Let the measure of its complement be \({x^ \circ }\) Then,
\(x+60=90 \Rightarrow x=(90-60)=30\)
Therefore, the complement of the given angle is \({30^ \circ }\)

Q.2. Find the supplement of the given angle: 125° 
Ans: The given angle measures \({125^ \circ }\)
Let its supplement be \({x^ \circ }\) Then,
\(x+125=180 \Rightarrow x=(180-125)=55\)
Therefore, the supplement of the given angle measures is \({55^ \circ }\)

Q.3. In the adjoining figure, what value of x will make AOB a straight line?
Ans: \(AOB\) will be a straight line if \(\angle A O C+\angle B O C=180^{\circ}\)
Therefore, \((3 x+5)+(2 x-25)=180 \Rightarrow 5 x-20=180\)
\(\Rightarrow 5 x=(180+20)\)
\(\Rightarrow 5 x=200\)
\(\Rightarrow x=40\)
Hence, \(x=40\) will make \(AOB\) a straight line.

Q.4. Find the angle, which is its complement.
Ans: Let the measure of the obtained angle be \({x^ \circ }\) Then,
\(x+x=90\)
\(\Rightarrow 2 x=90\)
\(\Rightarrow x=45\)
Hence, the required angle measures \({45^ \circ }\)

Q.5. Find the measure of an angle that is the complement of itself?
Ans: Let the measure of the angle be \({x^ \circ }\)
Then, the measure of its complement is given to be \({x^ \circ }\).
Since the sum of the measure of an angle and its complement is \({90^ \circ }\)
\(\therefore {x^ \circ } + {x^ \circ } = {90^ \circ }\)
\(\Rightarrow 2{x^ \circ } = {90^ \circ }\)
\(\Rightarrow {x^ \circ } = {45^ \circ }\)
Hence, the measure of the angle is \({45^ \circ }\)

Q.6. Find the angle which is equal to its supplement.
Ans: Let the measure of the angle be \({x^ \circ }\)
Then, the measure of its supplement \({=x^ \circ }\)
But, the measure of an angle \(+\) measure of its supplement \(= {180^ \circ }\)
\(\therefore {x^ \circ } + {x^ \circ } = {180^ \circ }\)
\( \Rightarrow 2{x^ \circ } = {180^ \circ }\)
\(\Rightarrow {x^ \circ } = {90^ \circ }\)
Hence, the measure of the required angle is \({90^ \circ }\)

Q.7. Two supplementary angles differ by 34°. Find the angles.
Ans: Let an angle be \({x^ \circ }\)
Then, the other angle is \({(x + 34)^ \circ }\)
Now, \({x^ \circ }\) and \({(x + 34)^ \circ }\) are supplementary angles.
\(\therefore x + (x + 34) = 180\)
\(\Rightarrow 2 x+34=180\)
\(\Rightarrow 2 x=180-34\)
\(\Rightarrow 2 x=180-34\)
\(\Rightarrow x=73\)
Hence, the measure of two angles is \({73^ \circ }\) and \({73^ \circ } + {34^ \circ } = {107^ \circ }\)

Q.8. An angle is equal to five times its complement. Determine its measure.
Ans: Let the measure of the given angle be \(x\) degrees. Then, its complement is \((90-x).\)
It is given that Angle \(=5×\) complement of the angle.
\(\therefore x + 5\left( {90 – x} \right)\)
\(\Rightarrow x=450-5 x\)
\(\Rightarrow 6 x=450\)
\(\Rightarrow x=75\)
Hence, the measure of the given angle is \({75^ \circ }\)

Summary 

In the given article, we discussed the pairs of angles, including linear pairs of angles, vertically opposite angles. Then we talked about the pair of supplementary angles and examples and then discussed the pair of complementary angles. We have provided some of the solved examples along with a few FAQs.

FAQs

Q.1. What are the types of angle pairs?
Ans: The four types of angle pairs are given below:
1. Adjacent angle
2. Vertically opposite angles
3. Complementary angles
4. Supplements angles

Q.2. What are the four different pairs of angles?
Ans: In geometry, we have different pairs of angles, and they are written below:
1. Complementary angles
2. Supplementary angles
3. Vertically opposite angles
4. Linear pairs
5. Adjacent angles

Q.3. What are angles and pairs of angles?
Ans: Angle: An angle is formed when the two rays originate from the same originating point. The rays making an angle are known as the arms of the angle, and the originating point is known as the vertex of the angle.
Pair of angles: In geometry, we often come across pairs of angles that have been given specific names.
Adjacent angles: Two angles in a plane are known as the adjacent angles if
a) they have a common vertex.
b) they have a common arm, and 
c) their other arms lie on the opposite sides of the common arm.

Q.4. What type of angle pair are 1 and 3?

Ans: These angles always have the same measure, so they are known as vertical angles. Hence, \(\angle 1\) and \(\angle 3\) are vertical angles.

Q.5. What is a linear pair example?
Ans: A linear pair is the pair of the adjacent angles that are formed when the two lines intersect. Like, assume \(\angle 1\) and \(\angle 2\) forms the linear pair. So do \(\angle 2\) and \(\angle 3,\angle 3\) and \(\angle 4\) and \(\angle 1\) and \(\angle 4\)

Learn about different Types of Angles here

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