as Paul Andrews suggests, in his fine article (reprinted by nrich), measuring angle tasks can be dull, even where students have estimated their sizes first

Paul suggests drawing triangles connecting dots on various point-circles

5, 6 and 9 dot circles produce angles that are all integer

students can be asked to draw all the different triangles they can find on e.g. 9 dot circles

and then measure the three angles

they need to draw these as accurately as they can

although it isn't intended to be part of this task, some students might note that equal angles subtend equal arcs

there are:

2 different triangles for a 5 point circle with angles that are M(36)

3 different triangles for a 6 point circle with angles that are M(30)

7 different triangles for a 9 point circle with angles that are M(20)

the circles are drawn fairly large to aid accurate measurement with a protractor:

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