A central angle equals the arc it opens. An inscribed angle is half that arc. The vertex tells you which one you have. Circle measure in general is in circle problems.
Where the vertex sits
A central angle has its vertex at the center. If the arc is 80°, the central angle is 80°. An inscribed angle has its vertex on the circle. The same 80° arc gives an inscribed angle of 40°. An angle inscribed in a semicircle is 90°, because it sees an arc of 180°.
Arc length and sector area use the central angle, not the inscribed one. Those formulas are in arc length and sector area.
Do not treat every angle in a diagram as central
If the vertex is on the circumference, halve the arc. If it is at the center, do not. Angle relationships that are not inside a circle are in lines and angles. The formula list is in ACT Math formulas.
The bottom line
Center: angle equals arc. On the circle: angle is half the arc.
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