"The definition of a good mathematical problem is the mathematics it generates rather than the problem itself." — Andrew Wiles from an interview for PBS website on the NOVA program, 'The Proof'.
The following problem is generalized from a question in Mathematics Paper 2 of the 1949 National Senior Examinations for the Union of South Africa.
Given any trapezium ABCD with AB // DC and line EF constructed as indicated so that ∠AEF = ∠DCF. What do you notice about angles BEC and AFD?
Explore your observation by dragging any of A, B, C, D or E. Can you deductively explain (prove) why your observation is true?
Is the result still valid if E is dragged 'outside' AD onto its extension? What if point D is dragged towards C, and then past C? Is the result still valid? Can you deductively explain your observations?
Matric Exam Geometry Problem - 1949Further exploration
1) Can you formulate alternative, equivalent versions of the result?
2) When, or rather where, is ∠BEC a maximum as E is dragged along line AD? Can you determine the optimal position?
Hint: If you get stuck with 2) above, go to this dynamic sketch "Determining maximum angle", or to check your solution.
Check your explorations above, by reading my 2015 paper in At Right Angles and Learning and Teaching Mathematics at "Flashback to the Past: a 1949 Matric Geometry Question".
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Created by Michael de Villiers, 13 August 2015.