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: Crossed Quadrilateral Angle Sum.
"... definitions are frequently proposed and argued about when counterexamples emerge ..." - Lakatos (1976:16)
The main purpose of the activity below is to actively engage students and learners in a Lakatosian experience by providing them with a counter-example to the quadrilateral angle sum result when the given quadrilateral ABCD is dragged into the shape of 'crossed quadrilateral' (De Villiers, 2004). Such an experience naturally leads to the careful re-examination of the concepts of 'quadrilateral', 'interior angle', etc. and provides a rich context for introducing students to more advanced mathematical concepts such as 'directed angles', 'directed distances' and 'directed areas' (De Villiers, 2020).
Prerequisite: It is assumed below that learners or students have already completed the Quadrilateral Angle Sum activity.
Crossed Quadrilateral Angle Sum
Notes
1) Follow the instructions of the accompanying guided worksheet.
2) To construct a diagonal in Question 3, use the 'Segment' tool on the left.
3) Clicking on the 'Link to alternative approach' button to navigate to a new sketch.
4) In this new sketch, directed angles are used with counter-clockwise angles measured as positive and clockwise angles measured as negative. What do you now notice about the sum of the angles of a crossed quadrilateral in this case?
5) Now drag ABCD until it becomes concave. What do you notice about the sum of the angles in this case? Can you explain your observations?
Note: Directed angles as in 3) above are also used in exploring the sum of the alternate angles of a crossed cyclic hexagon at: Cyclic Hexagon Alternate Angles Sum (click on the appropriate link in the sketch).
Explore More
To explore the sum of the angles of convex, concave or crossed polygons in general, go the link given below.
Investigating a general formula for the interior angle sum of polygons
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2004). The Role and Function of Quasi-empirical Methods in Mathematics. Canadian Journal of Science, Mathematics and Technology Education, July, pp. 397-418.
De Villiers, M. (2015). I have a dream: Crossed Quadrilaterals - a Missed Lakatosian Opportunity?. Philosophy of Mathematics Education Journal, July, no. 29.
De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. Learning and Teaching Mathematics, Dec, no. 29, pp. 30-34.
Lakatos, I. (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge: Cambridge University Press.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Some Related Links
Triangle Angle Sum (Rethinking Proof activity)
Quadrilateral Angle Sum (Rethinking Proof activity)
Crossed Quadrilateral Properties
Investigating a general formula for the interior angle sum of polygons
Interior angle sum of convex, concave & crossed polygons: a general formula
Cyclic Hexagon Alternate Angles Sum
A generalization of the Cyclic Quadrilateral Angle Sum theorem
Varignon Area (Rethinking Proof activity)
Slaying a Geometrical Monster:
Finding the Area of a Crossed Quadrilateral (PDF)
Some External Links
Internal and external angles (Wikipedia)
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.