No, they might just be similar. Scaling is important. Is that a typo?

Since this was translated from Russian, I believe there is a typo (the quadrilaterals can obviously be similar, as previously pointed out).

parmenides51 wrote: The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily congruent ? indeed there is a typo, the last word is similar and not congruent , thanks for noticing

parmenides51 wrote: The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar? There is a counterexample when the qudrilateral can be concave. In the picture, $H$ is the orthocenter of $\triangle ABC$. $E$ is an arbitrary point on $DH$ , and $F$ is the point on $AC$ with $EF \parallel CH$. quadrilateral $ABHC $and $ABEF$ has same angles, and both has vertical diagonals. ($\because E$ is the orthocenter of $\triangle ABF$)

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