Alternate Interior Angles Theorem
Pronunciation: /ˈɔl.tər.nɪt ɪnˈtɪər.i.ər ˈæŋ.gəlz ˈθɪər.əm/ Explain

Figure 2: Transversal.


Proof of the Alternate Interior Angles Theorem
Step  Description  Justification 
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 

References
 McAdams, David E.. All Math Words Dictionary, Alternate Interior Angles Theorem. 2nd Classroom edition 201501084799968. pg 13. Life is a Story Problem LLC. January 8, 2015. Buy the book
Cite this article as:
McAdams, David E. Alternate Interior Angles Theorem. 4/11/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/a/altinterioranglestheorem.html.
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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.)
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.)