Consider the circle below with with the quadrilateral \(ABCD\) incribed inside:
Let point \(O\) be the center of the circle. Let \(\alpha = \angle BAD\), \(\beta = \angle BCD\), \(\theta = \angle BOD \ (\text{smaller})\), and let \(\epsilon = \angle BOD \ (\text{larger})\):
According to the inscribed angle theorem:
\[\begin{gather} \theta = 2 \alpha \\ \epsilon = 2 \beta \end{gather} \]
Since \(\theta + \epsilon = 360\), then:
\[\begin{gather} \theta + \epsilon = 2 \alpha + 2\beta = 360 \\ \alpha + \beta = 180 \end{gather} \]