One can find the theorems concerning inscribed angles in Euclid's Elements which would explain why Ptolemy assumes that they are known. They are Propositions XX through XXII of Book IV and roughly state the following.
Prop. XX The central angle of a circle is double the angle at the circumference.
Prop. XXI Inscribed angles in same segment of a circle are equal.
Prop. XXII The sum of opposite angles of a quadrilateral inscribed in a circle is two right angles.
One also wonders how he arrived at a value for the chord of 120° which is the side of a pentagon. The Pythagoreans knew how to construct a pentagon from a isosceles triangle with base angles of 72° and a peak angle of 36° (see Elements, Book IV, Prop. XI). But that doesn't tell how Ptolemy arrived at a magnitude for the side of the pentagon. One possibility is the use of the golden ratio which is found in Euclid's Elements (Book VI, Prop. XXX) and what is known as Ptolemy's theorem or some other method to find the chord of half the angle if the chord of an angle is known.
Ptolemy was not the first to construct a table of chords. The credit for that goes to Hipparchus who constructed a 7 1/2° chord table which Ptolemy later improved on with a 1/2° table. Hipparchus anticipated much of what Ptolemy did concerning astronomy. He is believed to have traveled to Alexandria. Hipparchus made astronomical observations at Rhodes where the Antikythera mechanism is believed to have come from.