Opposite Angle Congruence

Pronunciation: /ˈɒp ə sɪt ˈæŋ gəl ˈkɒŋ gru əns/ ?

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Manipulative 1: Opposite angle congruence. Created with GeoGebra.

Opposite angle congruence is a property of triangles.[1] If two angles of a triangle are equal, the sides opposite the equal angles are also equal. This property is sometimes called base angle congruence. In his book Elements, book 1 proposition 6, Euclid described this property.

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Proof

StepDiagramDescriptionJustification
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Let ΔABC be a triangle having the angle ∠ABC equal to the angle ∠ACB. The claim is that side AB is equal to side BC.

Initial conditions and statement of claim.

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If side AB is not equal to side AC, then one of them must be larger.

Euclid, Elements, Book 1, Common Notions, Translated and annotated by D. Joyce.

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Let segment AB be the larger. Put point D on AB such that DB is equal to AC.

Euclid. Elements Book 1 Proposition 3. Translated by D. Joyce. A line segment can be drawn in a larger line segment the size of a smaller line segment.
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Draw a line segment between points D and C.

Euclid. Elements, Book 1 Postulate 1 translated by D. Joyce: A line can be drawn using any two points.

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Since DB = AC, BC is in common and ∠ABC = ∠ACB by definition, triangles ΔABC = ΔDCB.

Euclid. Elements Book 1 Proposition 4: SAS Congruence.

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But, if ΔABC = ΔDCA, then line segment AC = DB which contradicts the assumption that AC > DB. So AC can not be greater than AB, and can not be less than AB, so it must be equal to AB.

Euclid. Elements, Book 1, Common Notions 5, translated by D. Joyce.

References

  1. Euclid. Elements, book 1, proposition 6. D. Joyce. Elements. (Accessed: 2009-12-21). http://babbage.clarku.edu/~djoyce/java/elements/bookI/propI6.html.

Cite this article as:


Opposite Angle Congruence. 2009-12-21. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/o/oppositeanglecongruence.html.

Translations

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Revision History


2009-12-21: Added "References" (McAdams, David.)
2008-11-13: Changed manipulatives to GeoGebra. Expanded article (McAdams, David.)
2007-09-10: Initial version (McAdams, David.)

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