বৃত্তস্থ চতুর্ভুজের কৌণিক ধর্মসমূহ

These facts are always true about a cyclic quadrilateral:

$1)$ Opposite angles always add up to 180 degrees.

(Briefly, this follows from the inscribed-angle-intercepted-arc relationship: we have $\angle{ADC}+\angle{ABC} = \frac12\widehat{ABC} + \frac12\widehat{ADC} = \pi$, where we've used the fact that the two arcs $\widehat{ABC}$ and $\widehat{ADC}$ combine to form the blue circle, whose circumference is $2\pi$.)

$2)$ If diagonals of cyclic quadrilateral $ABCD$ are drawn, then $\angle ACB = \angle ADB$, and the same is true for every similar pair of angles.

(Briefly, this follows from the same inscribed-angle-intercepted-arc relationship: the angles $\angle{ACB}$ and $\angle{ADB}$ are both half the measure of arc $\widehat{AB}$.)

$3)$ Both 1) and 2) are iff (if and only if) relationships, so you can also use them to prove that a quadrilateral is cyclic, giving you access to all the other properties; this makes cyclic quadrilaterals a VERY powerful tool in angle chasing problems. We will prove and investigate this more completely in a later topic.