When two parallel lines are intersected by a transversal the angles formed in each pair of interior angles are?

When two lines intersect they form two pairs of opposite angles, A + C and B + D. Another word for opposite angles are vertical angles.

Vertical angles are always congruent, which means that they are equal.

Adjacent angles are angles that come out of the same vertex. Adjacent angles share a common ray and do not overlap.

The size of the angle xzy in the picture above is the sum of the angles A and B.

Two angles are said to be complementary when the sum of the two angles is 90°.

Two angles are said to be supplementary when the sum of the two angles is 180°.

If we have two parallel lines and have a third line that crosses them as in the ficture below - the crossing line is called a transversal

When a transversal intersects with two parallel lines eight angles are produced.

The eight angles will together form four pairs of corresponding angles. Angles 1 and 5 constitutes one of the pairs. Corresponding angles are congruent. All angles that have the same position with regards to the parallel lines and the transversal are corresponding pairs e.g. 3 + 7, 4 + 8 and 2 + 6.

Angles that are in the area between the parallel lines like angle 2 and 8 above are called interior angles whereas the angles that are on the outside of the two parallel lines like 1 and 6 are called exterior angles.

Angles that are on the opposite sides of the transversal are called alternate angles e.g. 1 + 8.

All angles that are either exterior angles, interior angles, alternate angles or corresponding angles are all congruent.

Example

The picture above shows two parallel lines with a transversal. The angle 6 is 65°. Is there any other angle that also measures 65°?

6 and 8 are vertical angles and are thus congruent which means angle 8 is also 65°.

6 and 2 are corresponding angles and are thus congruent which means angle 2 is 65°.

6 and 4 are alternate exterior angles and thus congruent which means angle 4 is 65°.

Video lesson

Find the measure of all the angles in the figure

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.

View Answer >

go to slidego to slide

Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts

Book a Free Trial Class

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.