SAS Congruence

Pronunciation: /es eɪ es kən'gru əns/ Explain
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Manipulative 1: SAS Congruence. Created with GeoGebra.

Two triangles are congruent if two adjacent sides and the angle contained by the sides are congruent with corresponding sides and angle of the other triangle[1][2]. In this case we say that the triangles are SAS congruent. SAS stands for side, angle, side.

Proof of SAS Congruence

StepExampleDescriptionJustification
1 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Start with triangle ABC and DEF. Let AB be congruent to DE. Let BC be congruent to EF. Let ∠ABC be congruent to ∠DEF. Starting conditions.
2 blank space The claim is that AC ≅ DF, ∠BCA ≅ ∠EFD, ∠CAB ≅ ∠FDE, and ΔABC ≅ ΔDEF. Claim
3 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) If ΔABC is placed on top of ΔDEF, the point A is on top of point D. Also, the line segment AB is on top of DE, and the line segment BC is on top of DF. Since point A coincides with D and point C coincides with point F, then the line segment AC must coincide with the line segment DF. Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.
4 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Since all sides of ΔABC coincide with all sides of ΔDEF, ΔABC ≅ ΔDEF. It must also be true that ∠BCA ≅ ∠EFD and ∠CAB ≅ ∠FDE. QED. Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.

References

  1. Euclid. Elements, Book 1 Proposition 4. Translated by Joyce, D. Elements. Last Accessed 12/5/2009. http://babbage.clarku.edu/~djoyce/java/elements/bookI/propI4.html.
  2. Hilbert, David. The Foundations of Geometry. pg 10. Translated by Townsend, E. J., Ph. D.. The Open Court Publishing Company. 1950. Last Accessed 12/21/2009. http://www.gutenberg.org/files/17384/17384-pdf.pdf.

Cite this article as:

McAdams, David E. SAS Congruence. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/sascongruence.html.

Image Credits

Revision History

12/21/2009: Added "References". (McAdams, David E.)
11/1/2008: Changed manipulatives to GeoGebra. (McAdams, David E.)
8/25/2007: Initial version. (McAdams, David E.)

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