Alternate Interior Angles Theorem

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

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

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 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.

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:


Alternate Interior Angles Theorem. 2010-10-13. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/altinterioranglestheorem.html.

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Revision History


2010-10-13: Added proof (McAdams, David.)
2010-06-26: Initial version (McAdams, David.)

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