Alternate Interior Angles Theorem

Pronunciation: /ˈɔl tər nɪt ɪnˈtɪər i ər ˈæŋ gəlz ˈθɪər əm/ Explain

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

Two parallel lines labeled 'a' and 'b'. A transversal line intersects both parallel lines. Eight angles formed by the intersections of the transversal with the parallel lines are labeled. On the lower intersection, starting with the upper right angle and going clockwise, the intersections are labeled 1, 2, 3, and 4. On the upper intersection, starting with the upper right angle and going clockwise, the angles are labeled 5, 6, 7, and 8.
Figure 2: Transversal.

Proof of the Alternate Interior Angles Theorem
StepDescriptionJustification
1a || bThis is a criterion, a necessary condition.
2∠1 ≅ ∠5The Parallel Line Postulate states that, given a transversal of parallel lines, corresponding angles are congruent.
3∠5 ≅ ∠7The Vertical Angle Theorem states that vertical angles are congruent.
4∠1 ≅ ∠7The 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 ∠1 ≅ ∠5 and ∠5 ≅ ∠7, then ∠1 ≅ ∠7.
5∠2 ≅ ∠8
Q.E.D.
A similar argument can be made to show that ∠2 ≅ ∠8.
Table 1: Proof of the Alternate Interior Angles Theorem

Cite this article as:

McAdams, David E. Alternate Interior Angles Theorem. 7/10/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/altinterioranglestheorem.html.

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Revision History

6/13/2018: Removed broken links, changed Geogebra links to work with Geogebra 5, updated license, implemented new markup. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)

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