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Intermediate

The blockhat

Four congruent rectangles are placed in a hat-shaped configuration. What’s the angle between the lines connecting the opposite corners?

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Solution

The angle is right (90°).

Consider triangle AKN. It’s a right triangle for sure, but it is also isosceles since both AN and NK are the sum of a block length and width. Therefore, angle FAM is 45°.

From reflection symmetry in the vertical line through M it follows that angle AFM is 45° as well. From adding up angles in triangle AMF one infers that angle AMF is 90°. The required angle FMK is 180°-90°=90°.

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