6.1 Circle Vocabulary, Central & Inscribed Angles

Why Circles Matter on the Geometry Regents

Circles carry a 2-8% weight on the NYSED Regents Examination in Geometry, which usually means one or two Part I multiple-choice questions and, in some administrations, a short constructed-response part. The rules are compact and highly predictable, so mastering the vocabulary and the two core angle theorems is one of the fastest point-per-hour gains on the whole exam. This section teaches the New York Next Generation standards GEO-G.C.2 and GEO-G.C.3.

Core Vocabulary

Every circle problem is assembled from a small set of terms, and the Regents distractors are built to punish mix-ups. Learn each one precisely.

The full circle measures 360 degrees, a semicircle is 180 degrees, and the circumference is C = 2(pi)r = (pi)d. A minor arc measures under 180 degrees; a major arc measures over 180 degrees. Arcs also obey the arc addition postulate: adjacent arcs add, so the arcs around the full circle always total 360 degrees. That fact lets you chase a missing arc by subtraction.

Naming and Measuring Arcs

A minor arc is named by its two endpoints (for example, arc AB), while a major arc is named with three letters (arc ACB) so it is clear which way around the circle you travel. The measure of an arc is a number of degrees, not a length; do not confuse arc measure with arc length, which is covered in Section 6.3. One more foundational idea from standard GEO-G.C.1 is that all circles are similar: any circle can be mapped onto any other by a translation and a dilation, which is why every arc-and-angle rule scales without regard to a circle's radius.

The Central Angle Theorem

A central angle equals its intercepted arc. If a central angle measures 75 degrees, the minor arc it cuts off also measures 75 degrees. This one-to-one match lets you fill in arcs the moment any central angle is known, and vice versa.

The Inscribed Angle Theorem

An inscribed angle equals half its intercepted arc. If an inscribed angle intercepts a 120-degree arc, the angle measures 60 degrees; equivalently, the intercepted arc is twice the inscribed angle. This is the single most-tested circle fact on the Regents, so make it reflex.

Three corollaries follow directly and appear constantly:

• Semicircle corollary: an inscribed angle that intercepts a diameter (a 180-degree semicircle) is always a right angle, 90 degrees. When a triangle is inscribed in a circle with one side as the diameter, the opposite angle is 90 degrees.
• Same-arc corollary: two inscribed angles that intercept the same arc are congruent, no matter where their vertices sit on the major arc.
• Cyclic quadrilateral corollary: in a quadrilateral inscribed in a circle, opposite angles are supplementary and sum to 180 degrees.

Worked Example: Cyclic Quadrilateral

Quadrilateral ABCD is inscribed in a circle, angle A = (3x + 10) degrees and the opposite angle C = (2x + 20) degrees. Because opposite angles are supplementary:

(3x + 10) + (2x + 20) = 180, so 5x + 30 = 180 and x = 30.

Then angle A = 100 degrees and angle C = 80 degrees, which correctly sum to 180 degrees.

Worked Example: Same-Arc Corollary

Points A and B are fixed on a circle, and two different points C and D each lie on the major arc. Inscribed angles ACB and ADB both intercept the same arc AB. If arc AB measures 70 degrees, then both angles measure 35 degrees, even though the vertices C and D are in different places. Recognizing that two inscribed angles "looking at" the same chord are equal is a frequent Regents shortcut.

Worked Example: Congruent Chords and Arcs

In the same circle, congruent chords intercept congruent arcs (and lie equidistant from the center). If chord AB is congruent to chord CD, then arc AB is congruent to arc CD. On the Regents this becomes an equation: if arc AB = (4x + 12) degrees and arc CD = (7x - 33) degrees, set them equal, 4x + 12 = 7x - 33, giving 3x = 45 and x = 15.

Common Traps

• Central vs. inscribed: central angle = arc; inscribed angle = half the arc. Reversing these is the number-one error.
• Vertex location decides the rule: vertex at the center means central; vertex on the circle means inscribed. Identify the vertex first.
• Semicircle equals 90, not 180: the arc is 180 degrees, but the inscribed angle on it is 90 degrees.