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3.5: Alternate Interior Angles

  • Page ID
    2188
  • Angles on opposite sides of a transversal, but inside the lines it intersects.

    Alternate interior angles are two angles that are on the interior of \(l\) and \(m\), but on opposite sides of the transversal.

    f-d_b313250db66fc135445ed3dd3c50506c7d5bf3c4faf0bb904d13e994+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

    f-d_a3314f04c041c4e4e7cd77c12a9dadec3926b523462deb2726d9c47a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

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

    Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

    If

    f-d_7adc3b2e74fc08ba224a323728ef32dca9f0b0042fec18750610e3c4+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    then \(l\parallel m\).

    What if you were presented with two angles that are on the interior of two parallel lines cut by a transversal but on opposite sides of the transversal? How would you describe these angles and what could you conclude about their measures?

    For Examples \(\PageIndex{1}\) and \(\PageIndex{2}\), use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

    f-d_a9669d1041f370e9e54907e878108a83838d6cf9377d8d1f8aac5a10+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Example \(\PageIndex{1}\)

    \(\angle EAF\cong \angle FJI\)

    Solution

    None

    Example \(\PageIndex{2}\)

    \(\angle EFJ\cong \angle FJK\)

    Solution

    \(\overleftarrow{CG} \parallel \overleftarrow{HK}\)

    Example \(\PageIndex{3}\)

    Find the value of \(x\).

    f-d_ca38ab0710d052576dd2f77a26c9dbd478f26fe51964cfe9e67c547c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    The two given angles are alternate interior angles and equal.

    \(\begin{align*} (4x−10)^{\circ} &=58^{\circ}\\ 4x &=68 \\ x &=17 \end{align*}\)

    Example \(\PageIndex{4}\)

    True or false: alternate interior angles are always congruent.

    Solution

    This statement is false, but is a common misconception. Remember that alternate interior angles are only congruent when the lines are parallel.

    Example \(\PageIndex{5}\)

    The angles are alternate interior angles, and must be equal for \(a\parallel b\). Set the expressions equal to each other and solve.

    f-d_284856552bc18ed1a7273da249dbebe18087bcc5b077ae5ad4bb13d5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    \(\begin{align*} 3x+16^{\circ} &=5x−54^{\circ} \\ 70&=2x \\ 35 &=x\end{align*}\)

    To make \(a\parallel b\), \(x=35\).

    Review

    1. Is the angle pair \(\angle 6\) and \(\angle 3\) congruent, supplementary or neither?
      f-d_c433bbb430c5a21eef1034baabdc9ea175e53f1b202b64c654c172d5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{7}\)
    2. Give two examples of alternate interior angles in the diagram:
      f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)

    For 3-4, find the values of \(x\).

    1. f-d_ef9c9e5e1682147552c76009265b7367a902b7a14102006cb31b7352+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{9}\)
    2. f-d_87a68f688acd773c7556b7fdab4f683f42aae988779a21d5a14f4b73+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{10}\)

    For question 5, use the picture below. Find the value of \(x\).

    f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)
    1. \(m \angle 4=(5x−33)^{\circ}\), \(m \angle 5=(2x+60)^{\circ}\)
    1. Are lines \(l\) and \(m\) parallel? If yes, how do you know?
      f-d_3a6cd4f8211f54b79476e4ce49e6493d6df49a9901f99dfa35cbaa67+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{12}\)

    For 7-10, what does the value of \(x\) have to be to make the lines parallel?

    f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{13}\)
    1. \(m \angle 4=(3x−7)^{\circ}\) and \(m \angle 5=(5x−21)^{\circ}\)
    2. \(m \angle 3=(2x−1)^{\circ}\) and \(m \angle 6=(4x−11)^{\circ}\)
    3. \(m \angle 3=(5x−2)^{\circ}\) and \(m \angle 6=(3x)^{\circ}\)
    4. \(m \angle 4=(x−7)^{\circ}\) and \(m \angle 5=(5x−31)^{\circ}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.4.

    Resources

    Vocabulary

    Term Definition
    alternate interior angles Alternate interior angles are two angles that are on the interior of two different lines, but on the opposite sides of the transversal.
    alternate exterior angles Alternate exterior angles are two angles that are on the exterior of two different lines, but on the opposite sides of the transversal.

    Additional Resources

    Video: Alternate Interior Angles Principles - Basic

    Activities: Alternate Interior Angles Discussion Questions

    Study Aids: Angles and Transversals Study Guide

    Practice: Alternate Interior Angles

    Real World: Alternate Interior Angles

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