A square deal

A regular hexagon and a square share a vertex. What’s the angle α?

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The angle α is 45°.

First, mirror point F in line AJ to obtain triangle AJF’ congruent to AJF. Since ∠IF’A=∠JFA=120°, we see using ASS that traingle IF’A is congruent to triangle GBA. It is then easy to see that ∠F’AB is 90°. and so ∠F’AJ is 15°. It follows that α=45°.

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