Problem — Halves of supplementary angles are complementary

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Prove that the halves of two supplementary angles are complementary. Then apply the result to the case in which one of the two angles measures $124^\circ$: compute the sizes of the two halves and verify that they are complementary.

Solution

Let $\alpha$ and $\beta$ be two supplementary angles, that is $\alpha + \beta = 180^\circ.$ Dividing both sides by $2$ gives $\frac{\alpha}{2} + \frac{\beta}{2} = 90^\circ,$ so the two halves $\dfrac{\alpha}{2}$ and $\dfrac{\beta}{2}$ have a right angle as their sum: they are complementary. $\blacksquare$

Numerical case. If $\alpha = 124^\circ$, its supplement is $\beta = 180^\circ - 124^\circ = 56^\circ.$ The two halves are $\frac{\alpha}{2} = 62^\circ, \qquad \frac{\beta}{2} = 28^\circ,$ and indeed $62^\circ + 28^\circ = 90^\circ$.

$\boxed{\dfrac{\alpha}{2} = 62^\circ,\quad \dfrac{\beta}{2} = 28^\circ}$