Absolute geometry
Circular arc measure
An arc is the set of all points where the circle intersects a closed half plane. Arcs associated with a
circle:
• A minor arc PQR is the intersection of a central angle ∠P OR and its interior (point Q lies on
both the circle and in the interior of ∠P OR ).
• A major arc PSR is the intersection of a central angle ∠P OR and its exterior (point S lies on
both the circle and in the exterior of ∠P OR ).
• A semicircle is an arc formed by the intersection of a circle with a closed half plane whose edge
passes through the center O. The end points of the arc lie on the endpoints of a diameter.
An arc is the subtended (or intercepted) arc of an angle if it is the intersection of that angle and its
interior with the circle.
Let P , Q, R, S, T , U, V be points on the given circle centered at O.
PQR is subtended by central angle ∠P OR.
TUV is subtended by inscribed angle ∠T SV .
Arc measure
• The measure of a minor arc is defined to be equal to the measure of the central angle that
subtends it:
Let P , Q, and R be points on the given circle centered at O.
mPQR = m∠P OR .
In the example shown, the measure of both the arc and the central angle is 60.
• The measure of a semicircle is defined to be 180.
mPQR = 180. P and R are both endpoints of a diameter.
• The measure of a major arc is defined to be 360 - (the measure of the complementary minor
arc). Since major arcs are defined by the exteriors of central angles, if the measure of the central
angle is x, the measure of the major arc is 360 - x.
mPQR = 360 − m∠P OR.
In the example shown, the measure of the central angle is 60, so the measure of the
major arc formed by the intersection of its exterior and the circle is 360 - 60 = 300.
Theorem: additivity of arc measure
Suppose arcs A1 = APB and A2 = BQC are any two arcs of circle O having just one point B in
common and such that their union A1 ∪ A2 = ABC is also an arc. Then m(A1 ∪ A2 ) = mA1 + mA2 .
Example: Given m∠AOB = 60 and mABC = 110, what are the values of mBQC, mAPB, and
m∠BOC? Justify.
Solution:
Example:
Given A, B, C, P , Q, R points on a circle centered at O, ordering and betweenness as
apparent, A and B endpoints of a diameter of the circle, and mABC = 250, find
• m∠AOC
• mAPB
• mAQB
• mBQC
• mARC
• m∠BOC
Solution: See posted live solution.