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3.7: Same Side Interior Angles

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    Angles on the same side of a transversal and inside the lines it intersects.

    Same side interior angles are two angles that are on the same side of the transversal and on the interior of (between) the two lines.

    f-d_a8d9b91df19f74f34fa164377fd63a7dbc7c055fc5ff57ecbe9634bf+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.

    f-d_91ce9eaa5c4a427538d80d88915519d0a21cfbabb9534f1f08cb0850+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    If \(l \parallel m\), then \(m\angle 1+m\angle 2=180^{\circ}\).

    Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.

    If

    f-d_67d340685341a2f8c3028d57936483cd844c7c3451729fd0f1cfc6b0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    then \(l \parallel m\)

    Suppose you were presented with two angles that are on the same side of a transversal and between the two parallel lines crossed by the transversal. How would you describe these angles and what could you conclude about their measures?

    Example \(\PageIndex{1}\)

    Is \(l \parallel m\)? How do you know?

    Solution

    These angles are Same Side Interior Angles. So, if they add up to \(180^{\circ}\), then \(l\parallel m\).

    \(130^{\circ}+67^{\circ}=197^{\circ}\), therefore the lines are not parallel.

    f-d_74e7e9a69fe84ce8ccc252f406f9133f3718d58847748e2a3b59e5a1+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Example \(\PageIndex{2}\)

    Give two examples of same side interior angles in the diagram:

    f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    There are MANY examples of same side interior angles in the diagram. Two are \(\angle 6\) and \(\angle 10\), and \(\angle 8\) and \(\angle 12\).

    Example \(\PageIndex{3}\)

    Find the value of \(x\).

    f-d_44014a86cf753c8bea839356ae4c8b578df175e0f1e2aab4cdf96ccc+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    The given angles are same side interior angles. Because the lines are parallel, the angles add up to \(180^{\circ}\).

    \(\begin{align*}(2x+43)^{\circ}+(2x−3)^{\circ} &=180^{\circ} \\ (4x+40)^{\circ} &=180^{\circ} \\ 4x &=140 \\ x &=35\end{align*}\)

    Example \(\PageIndex{4}\)

    Find the value of \(y\).

    f-d_ef9c9e5e1682147552c76009265b7367a902b7a14102006cb31b7352+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{7}\)

    Solution

    \(y\) is a same side interior angle with the marked right angle. This means that \(90^{\circ}+y=180\) so \(y=90\).

    Example \(\PageIndex{5}\)

    Find the value of \(x\) if \(m\angle 3=(3x+12)^{\circ}\) and \(m\angle 5=(5x+8)^{\circ}\).

    f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    Solution

    These are same side interior angles, so set up an equation and solve for \(x\). Remember that same side interior angles add up to \(180^{\circ}\).

    \(\begin{align*} (3x+12)^{\circ}+(5x+8)^{\circ} &=180^{\circ} \\(8x+20)^{\circ} &=180^{\circ} \\8x &=160 \\ x &=20 \end{align*}\)

    Review

    For questions 1-2, use the diagram to determine if each angle pair is congruent, supplementary or neither.

    f-d_c433bbb430c5a21eef1034baabdc9ea175e53f1b202b64c654c172d5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)
    1. \(\angle 5\) and \(\angle 8\)
    2. \(\angle 2\) and \(\angle 3\)
    3. Are the lines parallel? Justify your answer.
      f-d_9ea9b140c5da22ccd6c69206ecf7887faa5b7828b1f90c387659b054+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{10}\)

    In 4-5, 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 \(\PageIndex{11}\)
    1. \(\angle AFD\) and \(\angle BDF\) are supplementary
    2. \(\angle DIJ\) and \(\angle FJI\) are supplementary

    For 6-8, what does the value of \(x\) have to be to make the lines parallel?

    f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{12}\)
    1. \(m\angle 3=(3x+25)^{\circ}\) and \(m\angle 5=(4x−55)^{\circ}\)
    2. \(m\angle 4=(2x+15)^{\circ}\) and \(m\angle 6=(3x−5)^{\circ}\)
    3. \(m\angle 3=(x+17)^{\circ}\) and \(m\angle 5=(3x−5)^{\circ}\)

    For 9-10, determine whether the statement is true or false.

    1. Same side interior angles are on the same side of the transversal.
    2. Same side interior angles are congruent when lines are parallel.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.6.

    Vocabulary

    Term Definition
    same side interior angles Same side interior angles are two angles that are on the same side of the transversal and on the interior of the two lines.
    supplementary angles Two angles that add up to \(180^{\circ}\).
    transversal A line that intersects two other lines.

    Additional Resource

    Interactive Element

    Video: Same Side Interior Angles Principles - Basic

    Activities: Same Side Interior Angles Discussion Questions

    Study Aids: Angles and Transversals Study Guide

    Practice: Same Side Interior Angles

    Real World: Alternate Exterior Angles

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