Side-Angle-Side Congruence
Pronunciation: /saɪd 'æŋ.gəl saɪd kən'gru.əns/ Explain
Abbreviation: SAS Congruence
Click on the blue points and drag them to change the figure
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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
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.
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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.
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Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.
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4
|
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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.
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Euclid Elements Book 1 Common Notion 4: Things which coincide one with another are equal.
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References
- 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
- 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/17384/17384-pdf.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. Side-Angle-Side Congruence. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://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.)