Ptolemy's Theorem corollary: Chord$(2\alpha+2\beta)=BC$ and more

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So I've got this problem that is making me go a little insane, I'm not sure if I'm just missing simple identities or what. The diagrams and the problem are shown below.

enter image description here enter image description here I hope someone can give insight into how to do these few problems. We are given Ptolemy's Theorem for quadrilaterals,

$AB\times CD + BC\times AD=120BD$

Where the diameter of the circle in which the quadrilateral is inscribed is $120$, in other words, $AC=120,$ in Ptolemy's words.

I've already tried a couple things, but I've gotten stuck going down the plug-in-and-see-what-happens rabbit hole.

Thanks!