SideAngleSide 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^{[1][2]}. In this case we say that the triangles are
SAS congruent.
SAS stands for side, angle, side.
Proof of SAS Congruence
Step  Example  Description  Justification 
1


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


The claim is that AC ≅ DF,
∠BCA ≅ ∠EFD,
∠CAB ≅ ∠FDE,
and ΔABC ≅ ΔDEF.

Claim

3


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


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
 Euclid. Elements, Book 1 Proposition 4. Book 1, proposition 4. Translated by Joyce, D. Clark University. Last Accessed 12/13/2018. https://mathcs.clarku.edu/~djoyce/elements/bookI/propI4.html. Buy the book
 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. http://www.gutenberg.org/files/17384/17384pdf.pdf. Buy the book
More Information
 Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.
Cite this article as:
McAdams, David E. SideAngleSide Congruence. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/sascongruence.html.
Image Credits
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.)