Final answer:
To calculate the measure of an inscribed angle or minor arc in a circle, note that the inscribed angle's measure is half of the arc it subtends, and the formula for the arc length involves the angle in degrees, the circle's radius, and π.
Step-by-step explanation:
To find the measure of an inscribed angle or a minor arc in a circle, certain properties of circle geometry need to be utilized. An inscribed angle in a circle is defined by two chords that have a common endpoint on the circle's circumference. This endpoint is the vertex of the angle. If the measure of an inscribed angle is given as 54 degrees, it subtends a minor arc that is exactly twice its measure, i.e., 108 degrees in the circle.
On the other hand, if a minor arc measures 140 degrees, without the context it's not clear whether this is referring to the minor arc's central angle, which would also be 140 degrees, or an inscribed angle, which would then be half of 140 degrees (and that would be 70 degrees). In general, for an angle measured in degrees, the arc length 'As' of a circle with radius 'r' can be calculated using the formula As = (Δθ/360) × 2πr, where 'Δθ' is the angle in degrees.
Remembering that one degree contains 60 minutes of arc and each minute contains 60 seconds of arc, precise measurements on circles or spheres, such as astronomical observations, can be made.