Linear pair of angles 

When two rays of line intersect at a point, a pair of linear angles is formed. Common practical examples of linear pair of angles include-

1. The angle formed by the pair of scissors
2. Angles formed by the hands of a clock
3. Angles formed by the slices of a pizza
4. The angle formed by a justice balance
5. Angle formed at the intersection of an electric T-pole

Types of linear pair of angles

1. Adjacent angles: - Angles formed at a common side, common vertex but no common interior point are adjacent angles.

   In the given image, angle 1 and angle 2 are adjacent angles, whereas angle PQR and angle 1 are not adjacent angles. It is because angle 1 and angle 2 share a common vertex, point Q. They also share a common side (ray of line generating from point Q).

2. Linear pair: - A pair of angles formed on a straight line are said to be a linear pair. The sum of angles of a linear pair is always 180 degrees.

   Consider the image as shown. Here angle 2 and angle 1 are said to be a linear pair because they are formed on the same line with a standard arm S and common vertex point Q. Also, the sum of angles 1 and 2 is equal to 180 degrees.

3. Vertical angles: - Commonly known as vertically opposite angles, these angles are equal and opposite formed at a common point. If the two rays of the line meet at a point, then the pair of opposite angles is said to be vertically opposite angles and equal.

   In the given image, angles 1 and 2 and 3 and 4 are vertically opposite angles.
   Note – Angle 3 and 1 are a linear pair and not vertically opposite angles. Therefore, the same criterion is applicable for angles 2 and 4.

4. Supplementary angles: - If the sum of angles at a point is 180 degrees, then the pair of angles are supplementary. The pair can be at a single point or line, or it can even be a pair placed separately whose sum is equal to 180 degrees.

   In the given figure, angles 1 and 2 are supplementary angles placed at a common point. Angles X and Y are also a pair of supplementary angles because the sum of their angles is equal to 180 degrees. Therefore, even though they are placed separately, they are called supplementary angles by the property of supplementary angles.

Example

Find angles 1 and 2 (as shown in the figure) if the difference in angles 1 and 2 is 70 degrees, given that these angles are a pair of linear angles.

Solution As given in the question, ∠1 - ∠2 = 70° ………… (i)
By virtue of the property of linear pair, ∠1 + ∠2 = 180° ………… (ii)
Adding (i) and (ii), we get,
2 ∠1 = 250°
We get, ∠1 = 125°

Putting the value of angle 1 in equation (i), we get,
125° - ∠2 = 70°
This gives, ∠2 = 55°

Check – The sum of angles of a linear pair must be 180°. This gives ∠1 + ∠2 = 125° + 55° = 180°.