Problem
Two right triangles which have equal hypotenuses and equal acute angles are congruent.
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Solution
Let $∠B=∠E=∠R$, $AC=DF$ and $∠C=∠F$.
Place $D$ on top of $A$, $F$ on top of $C$, and $E$ on the same side as $B$ with respect to $AC$.
Since $∠C=∠F$, $FE$ overlaps $CB$.
Then since $AB$ and $DE$ also share the point $A$ and are perpendicular to $CB (=FE)$, they overlap.
$$∴ \ ⊿ABC≡⊿DEF.$$
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Reference
Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.10