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Corresponding Angles: Definition, Theorem & Examples

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  • 0:00 Definitions
  • 2:10 Corresponding Angles Theorem
  • 2:50 Examples
  • 4:20 Lesson Summary
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Lesson Transcript
Instructor
Sarah Spitzig

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, you will learn how to identify corresponding angles. You will also learn how to use a theorem to find missing angles and solve everyday geometry problems.

Definitions

Before getting into the definition of corresponding angles, let's first go over a few basics about angles, transversal lines, and parallel lines.

An angle is formed when two rays, a line with one endpoint, meet at one point. This one point where two rays meet is called a vertex. The angle is formed by the distance between the two rays. Angles in geometry are often referred to using the < symbol, so angle A would be written as <A.

angle

A transversal line is a line that crosses or passes through two other lines. Sometimes the two other lines are parallel, and the transversal passes through both lines at the same angle. However, the two other lines do not have to be parallel in order for a transversal to cross them, as you can see here:

transversal

A straight angle, also called a flat angle, is formed by a straight line. The measure of this angle is 180 degrees. A straight angle can also be formed by two or more angles that sum to 180 degrees.

In the image on the right, <1 + <2 = 180.

straight angle

Parallel lines are two lines on a two-dimensional plane that never meet or cross. When a transversal passes through parallel lines, there are special properties about the angles that are formed that do not occur when the lines are not parallel. Notice the arrows on lines m and n towards the left. These arrows indicate that lines m and n are parallel.

parallel lines

Corresponding angles are formed when a transversal passes through two lines. The angles that are formed in the same position, in terms of the transversal, are corresponding angles.

corresponding angles

In this picture of a window pane, <a and <b are corresponding angles because they are in the same position. They are both above the parallel lines and to the right of the transversal.

real life CA

Corresponding Angles Theorem

The Corresponding Angles Theorem states:

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

A theorem is a proven statement or an accepted idea that has been shown to be true. The converse of this theorem, which is basically the opposite, is also a proven statement:

If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.

These theorems can be used to solve problems in geometry and to find missing information. The diagram shows which pairs of angles are equal and corresponding. Notice that the lines are parallel.

corresponding angles

Examples

Find the measure of the missing angles in the following diagram. Assume the lines are parallel.

CA example

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Additional Activities

Additional Examples

In the following additional examples, students will demonstrate their knowledge on corresponding angles and the Corresponding Angles Theorem by finding angle measurements and using the theorem to show if two lines are parallel or not. In completing the examples, students will solidify their knowledge of corresponding angles and understand that corresponding angles are equal if and only if the lines intersected by the transversal are parallel. The examples begin with basic computation and increase in difficulty as students progress.

Examples

1) In the image below, lines A and B are parallel. Find all of the indicated angle measurements.

2) In the image below, lines A and B are parallel. What are the angle measurements of the indicated corresponding angles?

3) Are the lines A and B parallel? How do you know?

4) Are the lines A and B parallel? How do you know?

Solutions

1) Since the lines A and B are parallel, we know that corresponding angles are congruent. So we have that g = 107 degrees. Since angle b and the angle labeled 107 degrees form a flat angle, we have that b = 180 - 107 = 73 degrees. Then, since b and e are corresponding angles, e = 73 degrees. Similarly, since a and b form a flat angle, a = 180-73 = 107 degrees and since c and the angle labeled 107 degrees form a flat angle, we have c = 180 - 107 = 73 degrees. Then, using corresponding angles, angle d = 107 degrees and angle f = 73 degrees.

2) Since the lines A and B are parallel, we know that corresponding angles are congruent. So we have 3x + 7 = 7x - 5. Subtracting 3x from both sides of the equation gives us 7 = 4x - 5. Adding 5 to both sides gives 12 = 4x, and dividing by 4 gives us 3 = x. But this is not the answer to the question; we need to find the measure of the angles. One angle is 3(3) + 7 = 9 + 7 = 16 degrees. The other angle should also be 16 degrees, and we will check it to verify our answer. We have 7(3) - 5 = 21 - 5 = 16 degrees.

3) The lines A and B are not parallel, even though they visually look parallel, because we have corresponding angles that are not congruent. 85 degrees is not equal to 87 degrees, so the lines cannot be parallel, by the Corresponding Angle Theorem.

4) If the lines are parallel, then the corresponding angles will be equal. The corresponding angles are equal if (4x - 50) / 10 = (3/10)x + 5. Multiplying both sides by 10, we have 4x - 50 = 3x + 50. Subtracting 3x from both sides gives x - 50 = 50. Adding 50 to both sides yields x = 100. But if we substitute this answer back into the expressions for our angle measurements, we find that the angle measurements are: (4(100) - 50) / 10 = 35 and (3/10)(10) + 5 = 35. So the lines are parallel if and only if the angles shown are 35 degrees. However, if the image is drawn to scale, the angles in question should be greater than 90 degrees. So, the lines are not parallel.

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