Problem
When the two sides $AB$ and $BC$ of $∠ABC$ are, respectively, parallel to the two sides $DE$ and $EF$ of $∠DEF$ in the same direction or in the opposite direction, then
$$∠ABC=∠DEF.$$
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Solution
(1) When $AB$ and $BC$ are, respectively, parallel to $DE$ and $EF$ in the same direction
If the intersection of $DE$ and $BC$ is $G$, then $∠DEF$ and $∠DGC$ are the corresponding angles, and since $BC∥EF$,
$$∠DEF=∠DGC.$$
Since $∠ABC$ and $∠DGC$ are the corresponding angles and $AB∥DE$,
$$∠ABC=∠DGC,$$
$$∴ \ ∠ABC=∠DEF.$$
(2) When $AB$ and $BC$ are, respectively, parallel to $DE$ and $EF$ or in the opposite direction
$∠ABC$ and $∠EGC$ are the corresponding angles, and since $AB∥DE$,
$$∠ABC=∠EGC.$$
Since $∠EGC$ and $∠DEF$ are the alternate angles and $BC∥EF$,
$$∠EGC=∠DEF,$$
$$∴ \ ∠ABC=∠DEF.$$
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Reference
Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.8