Angle-Side-Angle Congruence

Pronunciation: /ˈæŋ.gəl saɪd ˈæŋ.gəl kən'gru.əns/ Explain
Abbreviation: ASA

In Euclidean geometry, two triangles are known to be congruent if two corresponding angles and the side they contain are congruent. Two angles and the side are congruent if they have the same measure. This is called ASA congruence. ASA stands for angle, side, angle.

In figure 1, angles A and A' are congruent, angles B and B' are congruent and line segment AB is congruent with line segment A'B'.

Click on the blue points and drag them to change the figure. When you change the figure, what stays the same and what changes. Some things to look at are the length of the lines, the size of the angles, and the orientation.

Manipulative 1 - Angle-Side-Angle Congruence of Triangles Created with GeoGebra.

Proof of ASA Congruence

Step	Diagram	Discussion
Click on the blue points and drag them to change the figures.
1		Let ΔABC be any triangle. Let A'B'C' be a triangle such that that the angle A'B'C' is congruent with angle ABC, and angle B'C'A' is congruent with angle BCA and side a is congruent with side a'. Note that point D is used only to rotate the triangle.
2	!!!!!UNKNOWN REFERENCE APPLETGGBasacongruence02.ggb!!!!!	Construct point C'' on line segment AC such that AC'' = A'C'.
3		Angles BCC'' = BAC = B'A'C' (given). Segments BC = B'C'. So, triangle BCC'' = B'A'C'.
4

References

1. McAdams, David E.. All Math Words Dictionary, ASA congruence. 2nd Classroom edition 20150108-4799968. pg 21. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

• Congruent Triangles - Two angles and included side (ASA). https://www.mathopenref.com/congruentasa.html. Math Open Reference. 6/19/2018. www.mathopenref.com.
• Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.

Cite this article as:

McAdams, David E. Angle-Side-Angle Congruence. 1/10/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/asacongruence.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/15/2018: Removed broken links, changed Geogebra links to work with Geogebra 5, updated license, implemented new markup. (McAdams, David E.)
12/21/2009: Added "References". (McAdams, David E.)
6/14/2008: Changed manipulative from Geometer's Sketchpad to GeoGebra. (McAdams, David E.)
6/7/2008: Corrected link errors. (McAdams, David E.)
9/24/2007: Corrected angle 'V' to become angle 'B'. (McAdams, David E.)
9/3/2007: Initial version. (McAdams, David E.)