Problem
If you drop a perpendicular line $AD$ from the right-angled vertex $A$ of a rectangular triangle $ABC$ to the side $BC$,
$$∠C=∠BAD \qquad and \qquad ∠B=∠CAD.$$
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Solution
$△ABC$ and $△DBA$ share $∠B \ (=∠DBA)$, and
$$∠A=∠ADB=∠R.$$
$$∴ △ABC \sim △DBA,$$
$$∴ \ ∠C=∠BAD.$$
$△ABC$ and $△DAC$ share $∠C \ (=∠ACD)$, and
$$∠A=∠CDA=∠R.$$
$$∴ \ △ABC \sim △DAC,$$
$$∴ \ ∠B=∠CAD.$$
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Reference Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.28.