Pairof lines examples

“Linear” simply means “arranged along a straight line.” We know that a straight angle is an angle that measures $180^\circ$. It is called a straight angle because it appears as a straight line. Two angles formed along a straight line represent a linear pair of angles.

Here, the angles $\angle a$ and $\angle b$ form a linear pair of angles. Also, they are congruent since they measure 90 degrees each.

Line pairphantom Radiography

3. Angles $^\angle ABC$ and $^\angle DBC$ form a linear pair of angles. Find the measure of $\angle ABC$ in the following figure.

It is the most common mistake to confuse supplementary angles with linear pairs of angles due to similarity in their properties. However, these are two different terms. Let’s understand the difference.

Linepairs per mm radiology

Line pairradiology

Linear pairs of angles are not congruent. When the measure of each of the angles in a linear pair is $90^\circ$, a linear pair of angles are congruent.

Supplementary is one of the necessary conditions for angles to be a linear pair. Hence, linear pairs are always supplementary. A linear pair forms a straight angle that measures $180^\circ$.

In the diagram shown above, the rays OA and OB are the non-common arms of the angles $\angle AOC$ and $^\circ BOC$. Rays OA and OB form a straight line AB.

Line pairphantom

A linear pair are two adjacent angles that sum to $180^\circ$. On the other hand, complementary angles are the angles that sum up to $90^\circ$. Complementary angles need not be adjacent.

Observe that these angles have one common arm (OP), which makes them adjacent angles. Also, their non-common sides OA and OB are opposite rays.

Line pairresolution

Line pairmath

Vertical angles are the angles that are opposite angles formed when two lines intersect each other. They only share a common vertex. Vertical angles are always congruent.

Let’s learn two important axioms which are collectively termed as ‘linear pair axioms’. Axiom is a mathematical statement that is self-evident and accepted to be true.

The converse of this postulate is not true. It means that if two angles are supplementary, they do not necessarily form a linear pair of angles.

Angles in a linear pair are supplementary. So, if two angles are not supplementary, they are not a linear pair of angles.

In the diagram shown below, $\angle POA$ and $\angle POB$ form a linear pair of angles. They add up to $180^\circ$. In other words, they are supplementary.