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Pegboard Quads
Age 14 to 16
Challenge Level 
An article by Paul Andrews Angle Measurement: An Opportunity for Equity, provides the best notes for background to this problem.
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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
No Right Angle Here
Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.