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Angles in parallel lines

Corresponding angles

Corresponding angles

Here you will learn about corresponding angles including how to recognize corresponding angles and apply this understanding to solve problems.

Students will first learn about corresponding angles as part of geometry in 8 th grade and expand their learning in high school.

What are corresponding angles?

Corresponding angles occur when a transversal line crosses two parallel lines. The pairs of angles formed on the same side of the transversal and in the same relative position are called corresponding angles and are equal in size.

You can often spot corresponding angles by drawing an F shape.

Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8 – Geometry (8.G.A.5)
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

• High School – Geometry (HSG.CO.C.9)
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

How to find corresponding angles

In order to find corresponding angles:

1. Identify two parallel lines that are intersected by a transversal.
2. Locate the angles that lie on the same side of the transversal and on different parallel lines.
3. Recognize that these pairs of angles are corresponding angles and are congruent.

Corresponding angles examples

Example 1: find the corresponding angles

\overleftarrow{\,C}\overrightarrow{D\,} and \overleftarrow{\,E}\overrightarrow{H\,} are parallel lines.

Which angle is the corresponding angle to \angle ABC?

1. Identify two parallel lines that are intersected by a transversal.

CD and EH are parallel lines. Line AG is the transversal.

2Locate the angles that lie on the same side of the transversal and on different parallel lines.

There are many pairs of corresponding lines, but the question asks which angle is a corresponding angle to \angle ABC. \angle BFE is on the same side of the transversal and on a different parallel line than \angle ABC.

3Recognize that these pairs of angles are corresponding angles and are congruent.

\angle BFE and \angle ABC are corresponding angles.

Example 2: find pairs of corresponding angles

\overleftarrow{\,S}\overrightarrow{Y\,} and \overleftarrow{\,T}\overrightarrow{Z\,} are parallel lines.

Name all pairs of corresponding angles.

Identify two parallel lines that are intersected by a transversal.

Locate the angles that lie on the same side of the transversal and on different parallel lines.

Recognize that these pairs of angles are corresponding angles and are congruent.

Example 3: corresponding angles in a real-world object

Look at the road intersections shown below. Which two letters represent corresponding angles?

Identify two parallel lines that are intersected by a transversal.

Locate the angles that lie on the same side of the transversal and on different parallel lines.

Recognize that these pairs of angles are corresponding angles and are congruent.

How to find the missing angle with corresponding angles

In order to find the missing angle with corresponding angles:

1. Highlight the angle(s) that you already know.
2. Use corresponding angles to find a missing angle.
3. Use basic angle facts if needed to calculate other missing angles.

Note: Steps 2 and 3 may be done in either order and may need to be repeated. Step 3 may not always be required.

Example 4: corresponding angles

Highlight the angle(s) that you already know.

Use corresponding angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

Example 5: corresponding angles

Highlight the angle(s) that you already know.

Use corresponding angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

Example 6: corresponding angles with algebra

Highlight the angle(s) that you already know.

Use corresponding angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

Teaching tips for corresponding angles

• Provide examples on student worksheets of corresponding angles alongside other types of angles, such as alternate interior angles or alternate exterior angles. This allows students to compare and contrast different angle relationships.

• Relate corresponding angles to real-life scenarios, such as the angles formed by intersecting streets, railway tracks, or architectural structures. This can help students understand the practical significance of corresponding angles

Easy mistakes to make

• Mixing up angle facts
There are a lot of angle facts and it is easy to mistake alternate angles with corresponding angles. To prevent this from occurring, think about corresponding angles as being underneath the F shape.

• Using a protractor to measure an angle
Using a protractor to find angle measures can be problematic. Most diagrams are not to scale, so using a protractor will not result in a correct answer unless it is a coincidence.

• Alternate interior angles theorem
• Parallel angles
• Same side interior angles

Practice corresponding angles questions

1. Which two angles are corresponding angles?

JKL and \angle LKM

\angle JKO and \angle KON

\angle MKO and \angle QOP

\angle MKO and \angle LKM

The only pair of angles listed that are on the same side of the traversal and on different parallel lines are \angle MKO and \angle QOP.

2. Find the measure of angle \theta.

\theta=96^{\circ}

\theta=84^{\circ}

\theta=264^{\circ}

\theta=74^{\circ}

Using corresponding angles, you can see the angle 96^{\circ}.

You can then use angles on a straight line:

\theta=180-96=84^{\circ}

3. Find the measure of \theta.

\theta=72^{\circ}

\theta=36^{\circ}

\theta=108^{\circ}

\theta=144^{\circ}

Using corresponding angles, you can see the angle 72^{\circ}.

You can then use angles on a straight line:

\theta=180-72=108^{\circ}

4. Find the measure of angle \theta.

\theta=60^{\circ}

\theta=58^{\circ}

\theta=62^{\circ}

\theta=122^{\circ}

You start by calculating the missing angle in the triangle:

180-(62+60)=58

You can then use corresponding angles to see that \theta=58^{\circ}

5. Find the measure for angle \theta.

\theta=106^{\circ}

\theta=95^{\circ}

\theta=115^{\circ}

\theta=65^{\circ}

Using corresponding angles, you can see that \theta is corresponding to 106+11, therefore, \theta=106+11=115^{\circ}

6. By finding the measure of x, find the size of each of the corresponding angles labeled in the diagram.

45^{\circ} and 45^{\circ}

45^{\circ} and 135^{\circ}

30^{\circ} and 20^{\circ}

50^{\circ} and 50^{\circ}

3x and 5 x-30 are corresponding angles and so are equal.

Therefore you can write 3 x=5 x-30.

You can then solve this:

\begin{aligned}3x&=5x-30 \\\\ 0&=2x-30 \\\\ 30&=2x \\\\ 15&=x\end{aligned}

\begin{array}{l}3\times 15 = 45^{\circ} \\\\ 5\times 15-30 = 45^{\circ}\end{array}

Corresponding angles FAQs

What are corresponding angles?

Corresponding angles are angle pairs formed when a transversal intersects two parallel lines. Pairs of corresponding angles are located on the same side of the transversal but on different parallel lines, and they have equal measures.

What are the two types of corresponding angles?

1. Corresponding angles formed by parallel lines and transversals
2. Corresponding angles formed by non-parallel lines and transversals

What is the corresponding angle postulate?

The corresponding angle postulate states that when a transversal intersects two parallel lines, the corresponding angles formed are congruent (for example, ., they have equal measures).

What is the converse of corresponding angles theorem?

The converse of the corresponding angles theorem states that if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.

What is the difference between corresponding angles and consecutive interior angles?

Corresponding angles are found on the same side of the transversal but on different parallel lines and have equal measures. Consecutive interior angles are located on the same side of the transversal and inside the intersecting lines, and are supplementary angles, meaning they add up to 180 degrees.

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