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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? Pythagorean Theorem proof by dissection

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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.

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Why does it follow that because 4 and 4' corresponding sides are parallels then the necessary angles are congruent? As I am working trying to find the axioms in this proof it would be nice to know where this basic statement comes from. –  m.Os Jul 26 '13 at 18:15

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