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.

What are corresponding angles?

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.

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

There are four pairs of angles that lie on the same side of the transversal and on different parallel lines. All 4 are pairs of corresponding angles.

Example 3: corresponding angles in a real-world object

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

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

Find the measure of the missing angle \theta. Justify your answer.

Highlight the angle(s) that you already know.

Use corresponding angles to find a missing angle.

Here you can label the corresponding angle on the diagram as 75^{\circ}.

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

Here \theta is on a straight line with 75^{\circ}

So,

\begin{aligned}\theta&=180^{\circ}-75^{\circ} \\\\ \theta&=105^{\circ}\end{aligned}

Example 5: corresponding angles

Find the measure of the missing angle \theta. Justify your answer.

Highlight the angle(s) that you already know.

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

As the sum of angles in a triangle is 180^{\circ},

\begin{aligned}\theta&=180^{\circ}-\left(63^{\circ}+56^{\circ}\right) \\\\ \theta&=61^{\circ}\end{aligned}

Example 6: corresponding angles with algebra

Find the measure of the missing angle \theta. Justify your answer.

Highlight the angle(s) that you already know.

Here, you need to find the value for x to find the value of \theta.

Use corresponding angles to find a missing angle.

4 x-14 and 3 x+7 are corresponding angles. This allows you to find the value of x as the two angles are equal:

\begin{aligned}&4x-14=3 x+7 \\\\ &4x=3x+21 \\\\ &x=21^{\circ}\end{aligned}

As x=21^{\circ}, you have the two angles 4 x-14=3 x+7=70^{\circ}.

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

Here you can combine the fact that you have an isosceles triangle and the sum of angles on a straight line is 180^{\circ}.

As the triangle is isosceles with the base of the triangle on one parallel line, the angles on either side of the top vertex of the triangle are symmetrical (this can also be calculated using the alternate angle rule). These angles form a straight line. You can therefore find \theta \text{:}

\begin{aligned}\theta&=180-(70+70) \\\\ \theta&=40^{\circ}\end{aligned}

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.

Related angles in parallel lines lessons

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

The next lessons are

• 2D shapes
• Quadrilateral
• Triangles
• Area
• Symmetry
• Lines

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