Problem
Suppose that two line segments $AB$ and $CD$ are parallel to each other in the same direction.
Then, if the point $P$ is between $AB$ and $CD$,
$$∠ABP+∠CDP=∠BPD.$$
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Solution
Draw a straight line $PQ$ through the point $P$ and parallel to $AB$ and $CD$.
Then, since $AB∥PQ$, and $∠ABP$ and $∠BPQ$ are the alternate angles,
$$∠ABP=∠BPQ.$$
Since $CD∥PQ$, and $∠CDP$ and $∠DPQ$ are the alternate angles,
$$∠CDP=∠DPQ,$$
$$∴ \ ∠ABP+∠CDP=∠BPQ+∠DPQ.$$
Then,
$$∠BPQ+∠DPQ=∠BPD,$$
$$∴ \ ∠ABP+∠CDP=∠BPD.$$
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Reference
Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.9