Quadrilateral 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: Quadrilateral Angle Sum.

As advocated by the Van Hiele theory of learning geometry, having children do various concrete tessellations with congruent cardboard quadrilaterals 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). Preferably learners or students should already have completed the Triangle Angle Sum activity and/or know the 'sum of exterior angles' result for convex and concave polygons.

Quadrilateral Angle Sum

Notes
1) Although one could (as many teachers do) let learners or students discover the angle sum result of a quadrilateral 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.
2) With reference to Question 4 in the worksheet (see link at the top), note that the half-turns used in the tessellation produces parallel lines and zig-zags (saws), i.e. 'equal alternate angles'.
3) To measure alternate angles, use the 'Angle' tool on the left.
4) To answer Questions 7 and 8 in the accompanying worksheet, navigate to new sketches by clicking on the corresponding 'Link to' buttons.

Explore More
To explore the sum of the angles of a crossed quadrilateral, or the sum of the angles of convex, concave or crossed polygons, go the links given below.
Crossed Quadrilateral Angle Sum
Investigating a general formula for the interior angle sum of polygons

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.
Van Hiele, P.M. (1973). Begrip & Inzicht. Muusses: Purmerend.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

Some External Links
Internal and external angles (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, 17 August 2025.