Pythagorean theorem proof by dissection

Question

I have this proof of the Pythagorean theorem, but in the last two lines of the fourth paragraph I can't seem to find geometrically how the congruency between $1$, $2$, $3$, $4$ and $1'$, $2'$, $3'$, $4'$ follows only from the parallel explanation. Until now, I divided $FHOK$ and $YPQS$ in two triangles and proved by SAS axiom (I'm using Hilbert's work) that the triangles $QSP$ and $HKO$ are congruent, but can't see how to prove the remained triangles nor to find and easier proof using only the parallels information. Could somebody help me, please?

Answer 1

Draw a line from $M$ to $P$ and from $K$ to $J$. They've shown that triangles $KOJ$ and $MCP$ are congruent by $SAS$, and that triangles $KAJ$ and $MXP$ are congruent by $ASA$. The second congruence follows from the first, because we know that line segment $MP$ is congruent to line segment $KJ$. The fact that the necessary angles are congruent to use $ASA$ follows because all corresponding sides of $4$ and $4'$ are parallel. They conclude that $4$ and $4'$ are congruent.