If the opposite angles of inscribed quadrilateral are always supplementary, where did I lose the train of thoughts in this example?

Question

I embedded this diagram to make easier the understanding of my question.

Starting with the quadrilateral ABCD, the central angle ABC that subtends the arch formed by the inscribed angle X, according to the “Inscribed Angle Theorem”, should be exactly twice the value of X. Hence, we could consider the central angle of the arch AEC equal to 360° - 2x.

Changing a bit the perspective, if we consider the angle ADC the inscribed angle that subtends the arch AEC, and so the angle ADC should be equal to half of the angle AEC also according to the “Inscribed Angle Theorem”. Then we get that the value of angle ADC is equal to 180 – x.

Corroborating with this assumption, the double of the inscribed angle of the arch AEC plus the double of the inscribed angle of the arch AC should totalize 360°. And in fact, considering the double of the last found angle, ADC, plus the double of the angle AEC, we reach a total of 360°.

Now my question. The opposite sides of an inscribed quadrilateral should sum up 180°. When I add 180 – x to 2x, this results in 180° + x. Therefore, the value of x should be equal to 0°, which doesn’t make sense.

Where did I lose the train of thought!?

Thanks in advance

Answer 1

The mistake is that $ABCD$ is not cyclic and you can not say that $$\measuredangle ADC+\measuredangle ABC=180^{\circ}.$$

Answer 2

The only inscribed quadrilateral in your image is $ADCE$.