Corresponding angles are congruent or supplementary

A line that passes through two distinct points on two lines in the same plane is called a transversal. A transversal forms four pairs of corresponding angles. Corresponding angles are pairs of angles that lie on the same side of the transversal in matching corners. One of the angles in the pair is an exterior angle and one is an interior angle.

In the diagram below transversal l intersects lines m and n. ∠1 and ∠5 are a pair of corresponding angles. Similarly, ∠2 and ∠6, ∠3 and 7, and ∠4 and ∠8 are also corresponding angles.

Corresponding angles postulate

The corresponding angles postulate states that if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Parallel lines m and n are cut by transversal l above, forming four pairs of congruent, corresponding angles: ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ 7, and ∠4 ≅ ∠8.

The converse of the postulate is also true. If the corresponding angles of two lines cut by a transversal are congruent, then the lines are parallel.

Example:

Using corresponding angles and straight angles, find the measures of the angles formed by the intersection of parallel lines m and n cut by transversal l below.

Corresponding angles are two angles that lie in similar relative positions on the same side of a transversal or at each intersection. They are usually formed when two parallel or non-parallel lines are cut by a transversal.

Remember that a transversal is a line that intersects two or more lines.

In our illustration above, parallel lines a and b are cut by a transversal which as a result, formed 4 corresponding angles. For example, \angle 2 and \angle 6 are corresponding angles. Why? Because both angles are located in matching corners or corresponding positions on the right-hand side of the transversal. In other words, each angle is located above the line and to the right of the transversal.

Here are our corresponding angles (must be in pairs) from the diagram and their location.

• \angle \textbf{1} and \angle \textbf{5} – above the line, left of the transversal
• \angle \textbf{3} and \angle \textbf{7} – below the line, left of the transversal
• \angle \textbf{2} and \angle \textbf{6} – above the line, right of the transversal
• \angle \textbf{4} and \angle \textbf{8} – below the line, right of the transversal

There are a few things to remember when dealing with corresponding angles.

Corresponding Angles Postulate

When two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent or have the same measure.

Take for example in our diagram above, since \angle 1 and \angle 5 are corresponding angles, they are congruent. This also means that if \angle 1 measures 70^\circ then \angle 5 also measures 70^\circ . Therefore, \angle 1 \cong \angle 5.

On the other hand, if the transversal intersects with two non-parallel lines, the corresponding angles formed are not congruent and do not have a specific relationship to each other.

Hence, \angle a and \angle e are corresponding angles but are NOT congruent.

Example Problems Involving Corresponding Angles

Example 1: Identify the corresponding angles.

Here we have two parallel lines, lines k and g, that are cut by the transversal, t. Remember that corresponding angles are angles that are in similar positions on the same side of the transversal.

So the corresponding angles are:

• \angle 2 and \angle 1
• \angle 4 and \angle 3
• \angle 6 and \angle 5
• \angle 8 and \angle 7

Example 2: Name the pairs of corresponding angles and their location.

As you can see, the transversal cuts across two non-parallel lines forming 4 corresponding angles. Always remember that in this case, though the angles are located in corresponding positions relative to the two lines, they are not congruent.

Are all corresponding angles supplementary?

Are corresponding angles always equal?

What angles are corresponding angles?

Are corresponding sides congruent?