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This problem involves a simple construction which gives a surprising and useful result. Having confirmed the sizes of angles this knowledge can be applied to make images and polyhedra. Different routes to solution can also lead to useful discussions, including how well learners explain their reasoning and the elegance of their methods.
What do you know?
Spending time making triangles and feeling confident about their properties is a useful starting point. Finding triangles that might be congruent, cutting them out and testing the congruency by matching them can then lead to identifying why sides and angles might be equal.
See:
Paper Folding - Models of the Platonic Solids.
Can learners justify all the results used as they work?