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3.6: Alternate Exterior Angles

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    Angles on opposite sides of a transversal, but outside the lines it intersects.

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

    f-d_e0e1b24cabd3907ddb79e7f5a22391da41b338ac18504818807dfee3+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

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

    f-d_c0e20623c4579ab6d6cba701289c1c80107494240ba3825fcfa7adda+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

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

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

    If

    f-d_9c7ec17fb02a7141e2063fd5ea712f254b1356eb99032e1f7c1b5905+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    then \(l \parallel m\).

    What if you were presented with two angles that are on the exterior 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 following diagram:

    f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Example \(\PageIndex{1}\)

    Give an example of a pair of alternate exterior angles.

    Solution

    \(\angle 1\) and \(\angle 14\) (many other possibilities)

    Example \(\PageIndex{2}\)

    Give another example of a pair of alternate exterior angles.

    Solution

    \(\angle 2\) and \(\angle 13\) (many other possibilities, must be different than answer to Example 1)

    Example \(\PageIndex{3}\)

    Find the measure of each angle and the value of \)y\).

    f-d_b442f21414ee3d898b7ce8a5109c44ba344a3687b10599412089ed5b+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Solution

    The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

    \(\begin{align*} (3y+53)^{\circ} &=(7y−55)^{\circ} \\ 108 &=4y \\ 27 &=y \end{align*}\)

    If \(y=27, then each angle is \([3(27)+53]^{\circ}=134^{\circ}\).

    Example \(\PageIndex{4}\)

    The map below shows three roads in Julio’s town.

    f-d_3b7db739f102b41478b199f5e4f610ccbd88018373e4d5ce6e85dc47+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    Solution

    Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.

    The \(130^{\circ}\) angle and \angle a are alternate exterior angles. If \(m \angle a=130^{\circ}\), then the lines are parallel.

    \( \begin{align*} \angle a+40^{\circ} &=180^{\circ} & by\:the \:Linear \:Pair \:Postulate \\ \angle a&=140^{\circ} & \end{align*}\)

    \(140^{\circ}\neq 130^{\circ}\), so Franklin Way and Chavez Avenue are not parallel streets.

    Example \(\PageIndex{5}\)

    Which lines are parallel if \(\angle AFG\cong \angle IJM\)?

    f-d_a9669d1041f370e9e54907e878108a83838d6cf9377d8d1f8aac5a10+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    These two angles are alternate exterior angles so if they are congruent it means that \(\overleftrightarrow{CG}\parallel \overleftrightarrow{HK}\).

    Review

    1. Find the value of \(x\) if \(m \angle 1=(4x+35)^{\circ}\), \(m \angle8=(7x−40)^{\circ}\):
      f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{8}\)
    2. Are lines 1 and 2 parallel? Why or why not?
      f-d_9b8cfcfc792afb149f56f2ed620d3111c7c34146302914c254a0c500+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{9}\)

    For 3-6, what does the value of \(x\) have to be to make the lines parallel?

    f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)
    1. \(m \angle2=(8x)^{\circ}\) and \(m \angle7=(11x−36)^{\circ}\)
    2. \(m \angle1=(3x+5)^{\circ}\) and \(m \angle8=(4x−3)^{\circ}\)
    3. \(m \angle2=(6x−4)^{\circ}\) and \(m \angle7=(5x+10)^{\circ}\)
    4. \(m \angle1=(2x−5)^{\circ}\) and \(m \angle8=(x)^{\circ}\)

    For 7-10, determine whether the statement is true or false.

    1. Alternate exterior angles are always congruent.
    2. If alternate exterior angles are congruent then lines are parallel.
    3. Alternate exterior angles are on the interior of two lines.
    4. Alternate exterior angles are on opposite sides of the transversal.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 3.5.

    Resources

    Vocabulary

    Term Definition
    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 Resource

    Interactive Element

    Video: Alternate Exterior Angles Principles - Basic

    Activities: Alternate Exterior Angles Discussion Questions

    Study Aids: Angles and Transversals Study Guide

    Practice: Alternate Exterior Angles

    Real World: Alternate Exterior Angles

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