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

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.

• Corresponding angles are angles that are at the same location at each intersection. In the interactive, they are color-coded to match. What do you notice about the measures of corresponding angles?

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  Corresponding angles have equal measures.

• Alternate interior angles are angles that are on opposite sides of the transversal and between the two parallel lines.
  

  What do you notice about the measures of alternate interior angles?

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  Alternate interior angles have equal measures.
• Alternate exterior angles are angles that are on opposite sides of the transversal and outside the two parallel lines.
  

  What do you notice about the measures of alternate exterior angles?

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  Alternate exterior angles have equal measures.
• Consecutive interior angles are angles that are on the same side of the transversal and between the parallel lines.


What do you notice about the measures of consecutive interior angles?

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Consecutive interior angles have measures that add up to 180°. They are supplementary.
• Consecutive exterior angles are angles that are on the same side of the transversal and outside the two parallel lines.
  

  What do you notice about the measures of consecutive exterior angles?

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  Consecutive exterior angles have measures that add up to 180°. They are supplementary.

Pause and Reflect

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

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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.

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.

Practice

1. In the diagram below, p ∥ q. ∠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?
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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.

In the diagram below, transversal l crosses parallel lines b and c. Use this diagram to answer questions 2 and 3.


2. 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?
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   (3x − 5) + 70 = 180; Consecutive exterior angles.
3. 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?
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   5y = 70; alternate exterior angles
4. Determine the value of x and y.

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   Solve the equations from questions 2 and 3.
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   x = 38 1 over 3 1 3 ; y = 14