# Alternate Interior Angles Theorem

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

 Click on the blue points and drag them to change the figure. As the figure changes, what remains the same? Manipulative 1 - Alternate Interior Angles Created with GeoGebra.

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. Figure 2: Transversal.

Proof of the Alternate Interior Angles Theorem
StepDescriptionJustification
1 a || b This is a criterion, a necessary condition.
2 ∠1 ≅ ∠5 The Parallel Postulate states that, given a transversal of parallel lines, corresponding angles are congruent.
3 ∠5 ≅ ∠7 The Vertical Angle Congruence Theorem states that vertical angles are congruent.
4 ∠1 ≅ ∠7 The 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

1. McAdams, David E.. All Math Words Dictionary, Alternate Interior Angles Theorem. 2nd Classroom edition 20150108-4799968. pg 13. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Alternate Interior Angles Theorem. 4/11/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/altinterioranglestheorem.html.

### Revision History

4/11/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)