When two parallel lines are cut by a transversal What are the angles between the two parallel lines?

Two lines that are stretched into infinity and still never intersect are called coplanar lines and are said to be parallel lines. The symbol for "parallel to" is //.

If we have two lines (they don't have to be parallel) and have a third line that crosses them as in the figure below - the crossing line is called a transversal:

In the following figure:

If we draw to parallel lines and then draw a line transversal through them we will get eight different angles.

The eight angles will together form four pairs of corresponding angles. Angles F and B in the figure above constitutes one of the pairs. Corresponding angles are congruent if the two lines are parallel. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs.

Angles that are in the area between the parallel lines like angle H and C above are called interior angles whereas the angles that are on the outside of the two parallel lines like D and G are called exterior angles.

Angles that are on the opposite sides of the transversal are called alternate angles e.g. H and B.

Angles that share the same vertex and have a common ray, like angles G and F or C and B in the figure above are called adjacent angles. As in this case where the adjacent angles are formed by two lines intersecting we will get two pairs of adjacent angles (G + F and H + E) that are both supplementary.

Two angles that are opposite each other as D and B in the figure above are called vertical angles. Vertical angles are always congruent.

$$\angle A\; \angle F\; \angle G\; \angle D\;are\; exterior\; angles\\ \angle B\; \angle E\; \angle H\; \angle C\;are\; interior\; angles\\ \angle B\;and\; \angle E,\; \angle H\;and\; \angle C\;are\; consecutive\; interior\; angles\\ \angle A\;and\; \angle G,\; \angle F\;and\; \angle D\;are\; alternate\; exterior\; angles\\ \angle E\;and\; \angle C,\; \angle H\;and\; \angle B\;are\; alternate\;interior\; angles\\ \left.\begin{matrix} \angle A\;and\; \angle E,\; \angle C\;and\; \angle G\\ \angle D\;and\; \angle H,\; \angle F\;and\; \angle B\\ \end{matrix}\right\} \;are\; corresponding\; angles$$

Two lines are perpendicular if they intersect in a right angle. The axes of a coordinate plane is an example of two perpendicular lines.

In algebra 2 we have learnt how to find the slope of a line. Two parallel lines have always the same slope and two lines are perpendicular if the product of their slope is -1.

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Find the value of x in the following figure

When two parallel lines are cut by a transversal, the angles are supplementary.

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In Geometry, when any two parallel lines are cut by a transversal, many pairs of angles are formed. There is a relationship that exists between these pairs of angles. While some of them are congruent, the others are supplementary. Let us learn more about the angles formed when parallel lines are cut by a transversal.

What are Parallel Lines Cut by Transversal?

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. When any two parallel lines are intersected by a line (known as the transversal), the angles that are subsequently formed, have a relationship. The various pairs of angles that are formed on this intersection are Corresponding angles, Alternate Interior Angles, Alternate Exterior Angles and Consecutive Interior Angles. Observe the figure given below which shows two parallel lines 'a' and 'b' cut by a transversal 'l'.

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

When two parallel lines are intersected by a transversal, the corresponding angles have the same relative position. 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.

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.

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.

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.

Related Links

Check out the following links related to Parallel Lines Cut by Transversal.

• Parallel Lines
• Transversal

1. Example 1: Identify the corresponding angles in the figure which shows two parallel lines 'm' and 'n' cut by a transversal 't'.

   

   Solution: In the given figure, two parallel lines are cut by a transversal, and the corresponding angles in the figure are ∠1 and ∠3; and ∠2 and ∠5.

2. Example 2: Find the value of x in the given parallel lines 'a' and 'b', cut by a transversal 't'.

   

   Solution: The given parallel lines are cut by a transversal, therefore, the marked angles in the figure are the alternate interior angles which are equal in measure. This means, 8x - 4 = 60°, and 8x = 64, x = 8.

   Therefore, the value of x = 8.

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FAQs on Parallel Lines Cut by Transversal

Parallel lines are straight equidistant lines that lie on the same plane and never meet each other. A transversal is any line that intersects two straight lines at distinct points. When any two parallel lines are intersected by a transversal, various angles are formed. There is a relationship that exists between these pairs of angles.

What happens When Parallel Lines are Cut by a Transversal?

When any two parallel lines are cut by a transversal, there are various pairs of angles that are formed. These angles are corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

What are the Special Pairs of Angles Formed when Parallel Lines Cut by Transversal?

When parallel lines are cut by a transversal, there are 4 special types of angles that are formed - corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles. While the pairs of corresponding angles, alternate interior angles, alternate exterior angles are congruent, the pairs of consecutive interior angles are supplementary.

How to Calculate Angle Measures in Parallel Lines Cut by a Transversal?

The unknown angles can be easily calculated when two parallel lines are cut by a transversal. The following facts help in finding the unknown angles. When parallel lines are cut by a transversal,

When Two Parallel Lines are Cut by a Transversal, are the Corresponding Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the corresponding angles that are formed are congruent.

When Two Parallel Lines are Cut by a Transversal, are the Alternate Interior Angles Congruent?

Yes, when two parallel lines are intersected by a transversal, the alternate interior angles are congruent.