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The folded sheet

A rectangular sheet of paper is folded in such a way that two equal angles are formed as shown. What fraction of the resulting quadrilateral is shaded?

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Solution

The shaded fraction is 1/2. We provide a solution suggested by Bannu based on triangle congruency only.

Extend the lines AD and FB to meet in point C. Since triangle AFB and ACB are both right triangles with the same angles and sharing a common side, they are congruent.

So the shaded triangle has half the surface of the large triangle AFC. It remains to be shown that the surface of this large triangle equals that of the original quadrilateral AFED.

Examining the triangles BFE and BCD, one can see that they also are similar right triangles. Their hypotenuses being the same, as follows from the larger triangle congruency, they must be congruent, completing the solution.

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