Alternate Exterior Angles Theorem

Pronunciation: /ˈɔl tər nɪt ɪkˈstɪər i ər ˈæŋ gəlz ˈθɪər əm/ ?

The Alternate Exterior Angles Theorem states that if two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

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Manipulative 1: Alternate Exterior Angles Theorem. Click on the blue points and drag them to change the figure.

Proof of the Alternate Exterior Angles Theorem

Two horizontal parallel lines with transversal line intersecting both lines. The upper line is labeled 'a'. The lower line is labeled 'b'. The eight angles formed by the two intersections are numbered as follows. From the upper left of the lower line clockwise, angles 1, 2, 3, and 4. From the upper left of the higher line clockwise, 5, 6, 7, 8.
Figure 2: Transversal.

StepDescriptionJustification
1Let a || bThis is a criterion, a necessary condition.
2To show: ∠5 ≅ ∠3 and ∠4 ≅ ∠6.The goal is to show that alternate exterior angles are congruent.
3∠5 ≅ ∠1The Parallel Line Postulate states that, given a transversal of parallel lines, corresponding angles are congruent.
4∠1 ≅ ∠3The Vertical Angle Theorem states that vertical angles are congruent.
5∠5 ≅ ∠3The Transitivity of Congruence of Angles Theorem states that if two angles are congruent to the same angle, they are congruent to each other. In this case since ∠5 ≅ ∠1 and ∠1 ≅ ∠3, so ∠5 ≅ ∠3. A similar argument can be made to show that ∠4 ≅ ∠6. Q.E.D.
Table 1: Proof of the Alternate Exterior Angles Theorem

Cite this article as:


Alternate Exterior Angles Theorem. 2011-05-07. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/altexterioranglestheorem.html.

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Revision History


2011-05-07: Improved wording and layout in proof. (McAdams, David.)
2010-06-26: Initial version (McAdams, David.)

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