In this section, you will investigate relationships among angle pairs created by two parallel lines that are crossed by a transversal.

illustration of parallel lines with a transversal

Interactive exercise. Assistance may be required. Use the interactive below to investigate the five relationships of angles formed by parallel lines and a transversal. Click the image to open the interactive, and it will open in a new browser tab or window. Drag the blue points to change the transversal. As you do so, notice how the angle measured for the eight labeled angles change.

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Use the interactive to answer the questions below.

Pause and Reflect

What patterns do you see in angle pairs that are congruent?

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Check Your Answer

Corresponding angles, alternate interior angles, and alternate exterior angles are congruent. If the angles correspond or are on opposite sides of the transversal and in the same region (inside or outside the parallel lines), then they are congruent. Close Pop Up

What patterns do you see in angle pairs that are supplementary?

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Consecutive interior angles and consecutive exterior angles are supplementary. If the angles are on the same side of the transversal and the same region (inside or outside the parallel lines), then they are supplementary.Close Pop Up

Practice

  1. In the diagram below, pq. ∠1 and ∠2 are alternate interior angles. ∠1 and ∠4 are consecutive interior angles. Using these facts explain why ∠1 and ∠4 must be supplementary.
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    ∠2 and ∠4 are supplementary angles because they make a straight line. How does this help relate ∠1 and ∠4?Close Pop Up
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    ∠2 and ∠4 are supplementary angles because they make a straight line. This means that m∠2 + m∠4 = 180°. Since ∠1 and ∠2 are congruent, m∠1 = m∠2. Substitute m∠1 for m∠2 into the first equation, and m∠1 + m∠4 = 180°. Hence, ∠1 and ∠4 are supplementary.Close Pop Up
  2. In the diagram below, transversal l crosses parallel lines b and c. Use this diagram to answer questions 2 and 3.

  3. Write an equation that you can use to solve for x. Which relationship did you use?

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    Identify the angle marked with an expression containing x and an angle with a given measurement. What is the relationship between these two angles?Close Pop Up

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    (3x − 5) + 70 = 180; Consecutive exterior angles.Close Pop Up
  4. Write an equation that you can use to solve for y. Which relationship did you use?

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    Identify the angle marked with an expression containing y and an angle with a given measurement. What is the relationship between these two angles?Close Pop Up

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    5y = 70; alternate exterior anglesClose Pop Up
  5. Determine the value of x and y.

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    Solve the equations from questions 2 and 3.Close Pop Up

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    x = 38 1 over 3 1 3 ; y = 14Close Pop Up