ANGLE RELATIONSHIP

Parallel Lines Cut by Transversal

In geometry, lines are classified into several types such as parallel, perpendicular, intersecting and non-intersecting lines, etc. For non-intersecting lines, we can draw a special line called transversal that intersects these lines at different points.

PARALLEL LINES

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

TRANSVERSALS

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

Angles Formed by Parallel Lines Cut by Transversal

When parallel lines are cut by a transversal, four types of angles are formed. Observe the following figure to identify the different pairs of angles and their relationship. The figure shows two parallel lines 'a' and 'b' which are cut by a transversal 'l'.

CORRESPONDING ANGLES

Corresponding angles are the angles which are formed in matching corners or corresponding corners with the transversal when two parallel lines and the transversal. In the figure given above, the corresponding angles formed by the intersection of the transversal are:

∠1 and ∠5
∠2 and ∠6
∠3 and ∠7
∠4 and ∠8

It should be noted that the pair of corresponding angles are equal in measure, that is, ∠1 = ∠5, ∠2 = ∠6, ∠3 = ∠7, and ∠4 = ∠8

ALTERNATE INTERIOR ANGLES

Alternate interior angles are formed on the inside of two parallel lines which are intersected by a transversal. In the figure given above, there are two pairs of alternate interior angles.

∠3 and ∠6
∠4 and ∠5

It should be noted that the pair of alternate interior angles are equal in measure, that is, ∠3 = ∠6, and ∠4 = ∠5

ALTERNATE EXTERIOR ANGLES

When two parallel lines are cut by a transversal, the pairs of angles formed on either side of the transversal are named as alternate exterior angles. In the figure given above, there are two pairs of alternate exterior angles.

∠1 and ∠8
∠2 and ∠7

It should be noted that the pair of alternate exterior angles are equal in measure, that is, ∠1 = ∠8, and ∠2 = ∠7

CONSECUTIVE INTERIOR ANGLES

When two parallel lines are cut by a transversal, the pairs of angles formed on the inside of one side of the transversal are called consecutive interior angles or co-interior angles. In the given figure, there are two pairs of consecutive interior angles.

∠4 and ∠6
∠3 and ∠5

It should be noted that unlike the other pairs given above, the pair of consecutive interior angles are supplementary, that is, ∠4 + ∠6 = 180°, and ∠3 + ∠5 = 180°.

Properties of Parallel Lines Cut by Transversal

When any two parallel lines are cut by a transversal they acquire some properties. In other words, any two lines can be termed as parallel lines if the following conditions related to the angles are fulfilled.

Any two lines that are intersected by a transversal are said to be parallel if the corresponding angles are equal.
Any two lines that are intersected by a transversal are said to be parallel if the alternate interior angles are equal.
Any two lines that are intersected by a transversal are said to be parallel if the alternate exterior angles are equal.
Any two lines that are intersected by a transversal are said to be parallel if the consecutive interior angles are supplementary.

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