Linear Pair Of Angles

We already know what angles are, Let us now focus on Linear pair of angles.

When the angle between two lines is 180°, they form a straight angle. A straight angle is just another way to represent a straight line. A straight line can be visualized as a circle with infinite radius. Line segment is any portion of a line which has two end points. Also, a portion of any line with only one end point is called a ray. A line segment with \(A\) and \(B\) as two end points is represented as \(\overline{AB}\). The figure shown below represents a line segment \(\overline{AB}\) and the two arrows at the end indicate a line.

Linear Pair

If a point O is taken anywhere on the line segment \(\overline{AB}\) as shown, then the angle between the two line segments AO and OB is a straight angle i.e. 180°.

Linear Pair

Consider a ray \(\overrightarrow{OP}\) stand on the line segment \overline{AB} as shown:

Linear Pair

The angles which are formed at O are ∠POB and ∠POA. It is known that the angle between the two line segments AO and OB is 180°. therefore, the angles ∠POB and ∠POA add up to 180°.

Thus, ∠POB + ∠POA = ∠AOB = \(180°\)

∠POB and ∠POA are adjacent to each other and when the sum of adjacent angles is 180° then such angles form linear pair of angles.

The above discussion can be stated as an axiom:

Axiom 1: If a ray stands on a line then the adjacent angles form a linear pair of angles.

Linear Pair

In the figure 2 given above, all the line segments pass through the point O as shown. As the ray \(\overrightarrow{OA}\) lies on the line segment \(\overline{CD}\),, angles ∠AOD and ∠AOC form a linear pair. Similarly, ∠QOD and ∠POD form a linear pair and so on.

The converse of the stated axiom is also true, which can also be stated as the following axiom:

Axiom 2: If two angles form a linear pair, then uncommon arms of both the angles form a straight line.

Linear Pair

Figure 3 Adjacent angles with different measures

In the figure shown above, only the last one represents a linear pair, as the sum of the adjacent angles is 180°. Therefore, AB represents a line. The other two pairs of angles are adjacent but they do not form a linear pair. They do not form a straight line.

The two axioms mentioned above form the Linear Pair Axioms and are very helpful in solving various mathematical problems. There is a lot more to learn about lines and angles. To know more about properties of pair of angles, download BYJU’s-The Learning App from Google Play Store.

Practise This Question

The angle between the bisectors of two adjacent supplementary angles is a right angle.