Problem — Complements of unequal acute angles

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Prove that if two acute angles are unequal, then their complements are unequal in the opposite sense: the larger of the two angles corresponds to the smaller complement.

Solution

Let $\alpha$ and $\beta$ be two acute angles with $\alpha > \beta$. Their complements are $\alpha' = 90^\circ - \alpha, \qquad \beta' = 90^\circ - \beta,$ and, since $\alpha,\beta$ are acute, both complements are positive angles.

From the inequality $\alpha > \beta$, changing signs gives $-\alpha < -\beta$; adding $90^\circ$ to both sides yields $90^\circ - \alpha < 90^\circ - \beta, \qquad \text{that is} \qquad \alpha' < \beta'.$

Hence the larger angle $\alpha$ corresponds to the smaller complement $\alpha'$: the complements are unequal in the opposite sense. $\blacksquare$