Problem
The opposite angles are equal. That is, when two straight lines $AB$ and $CD$ intersect at point $O$,
$∠AOC = ∠BOD$ and $∠AOD = ∠BOC$
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Solution
Since $COD$ is a straight line,
$$∠AOC+∠AOD = 2∠R.$$
Since $AOB$ is also a straight line,
$$∠AOD+∠BOD = 2∠R,$$
$$∴ \ AOC+∠AOD = ∠AOD+∠BOD.$$
$$∴ \ ∠AOC = ∠BOD.$$
Similarly,
$$∠AOD = ∠BOC.$$
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Reference
Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.5