Angle Relationships: Parallel Lines & Transversal

Two lines are said to be parallel when they do not intersect each other. We can also say, two lines that run along with each other and meet at infinity are called parallel lines.

What is Transversal?

When a line intersects two lines at distinct points, it is called a transversal. In the below figure, the line l intersects a and b at two distinct points P and Q.  Therefore, line l is the transversal line.

Lines cut by a transversal

In the below figure, the line c is not a transversal even if it intersects two lines a and b at O because it does not cut both of these lines at two distinct points. For a line to be transversal it must intersect two or more lines at separate points.

Intersecting Lines

Angle Relationship Between the Parallel Lines and Transversal

Various angle pairs are formed when a transversal intersects two or more parallel lines. Let us quickly recapitulate the angle relationships for the parallel lines cut by a transversal. Let a and b be two parallel lines intersected by the transversal l at the points P and Q as shown in the figure given below.

Transversal and Parallel lines

Based on the figure given above, we can create a table to show the pair of angles formed and their relationships.

Now, the question arises that how can we deduce if two lines are parallel or not? Is there any condition which demonstrates the parallel nature of two or more lines? Consider this situation.

Alan was asked by his teacher to draw two parallel lines. With the help of his set squares and ruler, he drew a straight line segment AB and then placed the set square on this line and drew two line segments XY and PQ by changing the position of the set squares as shown.

Interior Angle on the same side of transversal

He claimed that XY and PQ are parallel. Can you tell how? It’s simply right. The line segment AB serves as the transversal to XY and PQ and angle X and angle Y are corresponding angles in this case which are in equal measure. Therefore, XY and PQ are parallel line segments.

The following conclusions can be made:

1. If two lines are cut by a transversal such that the pair of corresponding angles are equal to each other, then the pair of lines are parallel.
2. If two lines are cut by a transversal such that the sum of interior angles on the same side of the transversal is supplementary, then the pair of lines are parallel.
3. If two lines are cut by a transversal such that the pair of alternate interior angles are equal to each other, then the pair of lines are parallel.

Thus, while checking if the two lines are parallel or not if any of the conditions mentioned above are satisfied, the lines are parallel to each other.

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