Cyclic Quadrilateral Alternate Angles Sum

The dynamic geometry activities below are from the "Proof as Explanation" section of 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: Cyclic Quadrilateral Worksheet & Teacher Notes.

Prerequisite
Though not essential, it is recommended to first complete the activity for four towns at Water Supply: Four Towns before completing the activity below.

Cyclic Quadrilateral Alternate Angles Sum

Notes
1) The main purpose of this activity is for students to discover and explain why (prove that) the alternate angles of a (convex) cyclic quadrilateral are supplementary & to generalize it to Duncan Gregory's theorem of 1836 for a (convex) cyclic 2n-gon.
2) Use the 'Calculate' tool on the left to sum the pairs of alternate (opposite) angles.
(If necessary, you can move the displayed measurements or calculations in the screen by clicking the arrow button on top of the item, and while holding it down, moving your selection to anywhere on the screen).
3) You might notice that if, for example, point D is dragged on the circumference of the circle, then ∠CDE remains unchanged. Can you explain why (prove) that is true?
Explore More
4) What about the converse? Is a (convex) quadrilateral necessarily cyclic if it has alternate (opposite) angles supplementary? Go to this dynamic sketch to explore this question: Cyclic Quadrilateral Converse.
5) For the Further Exploration of a Cyclic Hexagon as directed in the worksheet, click on the 'Link to Cyclic Hexagon' button to navigate to a new, blank sketch.
6) Using the toolbox on the left, follow the instructions in red in the new sketch to construct your own cyclic hexagon, and then measure & sum the two sets of alternate angles.
(To access the other tools in the toolbox like the Compass tool, Segment tool, Angle (measure) tool, use the vertical scroll bar to scroll down).
7) Alternatively to doing your own construction in 6), use this ready-made sketch of a cyclic hexagon at: Cyclic Hexagon Alternate Angles Sum.
8) To dynamically explore the sum of the alternate (opposite) angles of a crossed cyclic quadrilateral using directed angles, go to this sketch: Crossed Quadrilateral Properties and click on the 'Link to cyclic alternate angles sum' button.
9) For an additional investigation, you can also use the ready-made sketch in 7) to explore the possible angle sums of the alternate angles of a crossed cyclic hexagon using directed angles - click on the 'Link to crossed cyclic hexagon - directed angles' button. For more details, see De Villiers (2023).
Further Generalization
10) For a further generalization to cyclic 2n-gons (including some crossed cases) as discussed in De Villiers (1994), go here: A generalization of the Cyclic Quadrilateral Angle Sum theorem.

Related Circle Geometry Theorems
Apart from several related links given below, some readers may wish to explore the following novel proofs of familiar circle geometry theorems from high school:
Three Circle Geometry Theorem Proofs by Transformations

References
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2023). Investigating Alternate Angle Sums of Crossed Cyclic Hexagons. Learning & Teaching Mathematics, No. 35, pp. 28-31.
Mudaly, V. & Mahlaba, S.C. (2017). Teaching and learning cyclic quadrilateral theorems using Sketchpad in a Grade 11 class in South Africa. Ponte (International Journal of Sciences & Research), Vol. 73, No. 11, Nov, pp. 360-376. DOI: 10.21506/j.ponte.2017.11.23

Other Rethinking Proof Activities
Other Rethinking Proof Activities

Some External Links
Cyclic quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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, 30 June 2025; updated 1 July 2025; 9 Oct 2025; 25 Nov 2025; 26 Feb 2026.