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3.4: Corresponding Angles

  • Page ID
    2190
  • Identify corresponding angles as matching angles.

    Corresponding angles are two angles that are in the "same place" with respect to the transversal but on different lines. Imagine sliding the four angles formed with line \(l\) down to line \(m\). The angles which match up are corresponding.

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    Figure \(\PageIndex{1}\)

    Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the corresponding angles are congruent.

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    Figure \(\PageIndex{2}\)

    If \(l \parallel m\), then \(\angle 1\cong \angle 2\).

    Converse of Corresponding Angles Postulate: If corresponding angles are congruent when two lines are cut by a transversal, then the lines are parallel.

    If

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    Figure \(\PageIndex{3}\)

    then \(l \parallel m\).

    What if you were presented with two angles that are in the same place with respect to the transversal but on different lines? How would you describe these angles and what could you conclude about their measures?

    Example \(\PageIndex{1}\)

    If \(m\angle 2=76^{\circ}\), what is \(m\angle 6\)?

    f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    \(\angle 2\) and \(\angle 6\) are corresponding angles and l||m from the arrows in the figure. \(\angle 2\cong \angle 6\) by the Corresponding Angles Postulate, which means that \(m\angle 6=76^{\circ}\).

    Example \(\PageIndex{2}\)

    Using the measures of \angle 2 and \angle 6 from Example 1, find all the other angle measures.

    Solution

    If \(m\angle 2=76^{\circ}\), then \(m\angle 1=180^{\circ}−76^{\circ}=104^{\circ}\) (linear pair). \angle 3\cong \angle 2 \)(vertical angles), so \(m\angle 3=76^{\circ}\). \(m\angle 4=104^{\circ}\) (vertical angle with \angle 1\)).

    By the Corresponding Angles Postulate, we know \(\angle 1\cong \angle 5\), \(\angle 2\cong \angle 6\), \(\angle 3\cong \angle 7\), and \(\angle 4\cong \angle 8\), so \(m\angle 5=104^{\circ}\), \(m\angle 6=76^{\circ}\), \(m\angle 7=76^{\circ}\), and \(m\angle 8=104^{\circ}\).

    Example \(\PageIndex{3}\)

    Find the value of y:

    f-d_764c6382932c5ce076e8e9d394401386d1b5546376807aa3ce8ff988+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    The horizontal lines are marked parallel and the angle marked \(2y\) is corresponding to the angle marked 80 so these two angles are congruent. This means that \(2y=80\) and therefore \(y=40\).

    Example \(\PageIndex{4}\)

    If a||b, which pairs of angles are congruent by the Corresponding Angles Postulate?

    f-d_db29b4839d72e881911e626f7e7accc836a573e5d9a558caa2885019+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    There are 4 pairs of congruent corresponding angles:

    \(\angle 1\cong \angle 5\), \(\angle 2\cong \angle 6\), \(\angle 3\cong \angle 7\), and \(\angle 4\cong \angle 8\).

    Example \(\PageIndex{5}\)

    If \(m\angle 8=110^{\circ}\) and \(m\angle 4=110^{\circ}\), then what do we know about lines \(l\) and \(m\)?

    f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    \(\angle 8\) and \(\angle 4\) are corresponding angles. Since \(m\angle 8=m\angle 4\), we can conclude that l||m.

    Review

    1. Determine if the angle pair \(\angle 4\) and \(\angle 2\) is congruent, supplementary or neither:
      f-d_c433bbb430c5a21eef1034baabdc9ea175e53f1b202b64c654c172d5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    2. Give two examples of corresponding angles in the diagram:
      f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)
    3. Find the value of \(x\):
      f-d_11ab2e84cc5141c83bc06315adff74666e742c4301bf7cdc8e8e87c6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)
    4. Are the lines parallel? Why or why not?
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      Figure \(\PageIndex{11}\)
    5. Are the lines parallel? Justify your answer.
      f-d_19620e8aaf2620412ed42b0899b00aace1ba5472ca04b4405399e8d7+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)

    For 6-10, what does the value of \(x\) have to be to make the lines parallel?

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    Figure \(\PageIndex{13}\)
    1. If \(m\angle 1=(6x−5)^{\circ}\) and \(m\angle 5=(5x+7)^{\circ}\).
    2. If \(m\angle 2=(3x−4)^{\circ}\) and \(m\angle 6=(4x−10)^{\circ}\).
    3. If \(m\angle 3=(7x−5)^{\circ}\) and \(m\angle 7=(5x+11)^{\circ}\).
    4. If \(m\angle 4=(5x−5)^{\circ}\) and \(m\angle 8=(3x+15)^{\circ}\).
    5. If \(m\angle 2=(2x+4)^{\circ}\) and \(m\angle 6=(5x−2)^{\circ}\).

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.3.

    Resources

    Vocabulary

    Term Definition
    Corresponding Angles Corresponding angles are two angles that are in the same position with respect to the transversal, but on different lines.

    Additional Resource

    Interactive Element

    Video: Corresponding Angles Principles - Basic

    Activities: Corresponding Angles Discussion Questions

    Study Aids: Angles and Transversals Study Guide

    Practice: Corresponding Angles

    Real World: Corresponding Angles