Triangle Angle Sum

The dynamic geometry activity below is from my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Triangle Angle Sum.

As advocated by the Van Hiele theory of learning geometry, having children do various concrete tessellations with congruent cardboard triangles provides a rich conceptual context for discovering various geometric relationships (see Van Hiele, 1973; Fuys et al, 1984 & De Villiers, 1993/2010). Such hands-on activities with physical manipulatives can be nicely supplemented by dynamic geometry activities such as the one suggested below.

Prerequisites: It is assumed below that learners or students are already acquainted with some of the properties of parallel lines, specifically those of 'equal alternate angles' (zig-zag or saw)) and/or 'equal corresponding angles' (ladder).

Triangle Angle Sum

Notes
1) Although one could (as many teachers do) let learners or students discover the angle sum result of a triangle simply by measuring each angle and adding them with a calculator, such an approach, according to the van Hiele theory, does not provide an appropriate conceptual structure for the proof (explanation) that is to follow later. The activity suggested above, however, provides a more appropriate conceptual and visual framework (or scaffold) on which the formal explanation (proof) is later built.
2) To construct a line parallel to AB through point C as directed in the accompanying worksheet (see the Link at the top), use the 'Parallel' tool on the left.

Explore More
To explore the sum of the angles of a quadrilateral, or the sum of the angles on surfaces other than the Euclidean plane, go the links given below.
Quadrilateral Angle Sum
Crossed Quadrilateral Angle Sum
Spherical Triangle Angle Sum (Spherical Easel)
Hyperbolic Triangle Angle Sum (YouTube video)

References
De Villiers, M. (1993). Transformations: A golden thread in school mathematics. Spectrum, 31(4), 11-18.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2010). Some Reflections on the Van Hiele theory. Invited plenary presented at the 4th Congress of Teachers of Mathematics of the Croatian Mathematical Society, Zagreb, Croatia, 30 June –2 July 2010.
Fuys, D., Geddes, D. & Tischler, R. (Eds). (1984). English translation of selected writings of Dina van Hiele-Geldof and P. M. van Hiele. New York: Brooklyn College.
Fuys, D. & Geddes, D. (1984). An Investigation of Van Hiele Levels of Thinking in Geometry among Sixth and Ninth Graders: Research Findings and Implications. Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 27, 1984), 31.pp.
Guven, B. & Karataş, İ. (2009). Students discovering spherical geometry using dynamic geometry software. International Journal of Mathematical Education in Science and Technology, April, 40(3):331-340.
Lénárt, I. (Undated). Paper geometry vs Orange geometry – Comparative Geometry on the Plane and the Sphere.
Van Hiele, P.M. (1973). Begrip & Inzicht. Muusses: Purmerend.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

Some External Links
Euclid's Elements: Book I, Proposition 32 (David Joyce, Clark University)
Sum of angles of a triangle (Wikipedia)
Euclid's Elements of Geometry (by Heiberg, 1883-1885) (PDF)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Michael de Villiers, created with WebSketchpad, 16 August 2025.