Side-Angle-Side Congruence

Pronunciation: /saɪd 'æŋ.gəl saɪd kən'gru.əns/ Explain
Abbreviation: SAS Congruence
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Manipulative 1 - Side Angle Side 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[2][3]. In this case we say that the triangles are SAS congruent. SAS stands for side, angle, side.

Proof of SAS Congruence

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
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.
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.


  1. McAdams, David E.. All Math Words Dictionary, side-angle-side congruence. 2nd Classroom edition 20150108-4799968. pg 160. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Euclid. Elements, Book 1 Proposition 4. Book 1, proposition 4. Translated by Joyce, D. Clark University. Last Accessed 12/13/2018. Buy the book
  3. Hilbert, David. The Foundations of Geometry. pg 10. Translated by Townsend, E. J., Ph. D.. The Open Court Publishing Company. 1950. Last Accessed 12/13/2018. Buy the book

More Information

  • Euclid of Alexandria. Elements. Clark University. 9/6/2018.

Cite this article as:

McAdams, David E. Side-Angle-Side Congruence. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/5/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra app. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
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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