Geometry Reference Sheet Chapter 10 2016-2017
Definitions
Center
The given point from which all points on a circle are equidistant.
Central Angle
An angle whose vertex is the center of the circle.
Chord
A segment whose endpoints are on a circle.
Circle
The set of all points in a plane equidistant from a given point.
Circumscribed Angle
An angle whose sides are tangent to a circle.
Circumscribed Circle
The circle containing the vertices of an inscribed polygon.
Common Tangent
A line or segment that is tangent to two coplanar circles.
Concentric Circles
Coplanar circles that have a common center.
Congruent Arcs
Two arcs with the same measure that are arcs of the same circle or congruent circles.
Congruent Circles
Two circles for which a rigid motion or composition of rigid motions ma ps one circle onto the other.
Diameter
A chord that contains the center of a circle.
External Segment
The part of a secant segment that is outside the circle.
Inscribed Angle
An angle whose vertex is on the circle and whose sides contain chords of the circle.
Inscribed Polygon
A polygon is inscribed when all its vertices lie on a circle.
Intercepted Arc
An arc that lies between two lines, rays, or segments.
Major Arc (and
measure of)
The measure of a major arc is the difference between 360 and the measure of the related minor arc.
Minor Arc (and
measure of)
The measure of a minor arc is the measure of its central angle and is less than 180 .
Radius
A segment whose endpoints are the center and any point on a circle.
Secant
A line that intersects a circle in two points.
Secant Segment
A segment that contains a chord of a circle and has exactly one endpoint outside the circle.
Segments of the
Chord
When two chords intersect in the interior of a circle, each chord is divided into two segments that are
called the segments of the chord.
Semicircle (and
measure of)
An arc with endpoints that are endpoints of a diameter. Its measure is 180
Similar Arcs
Two arcs with the same measure.
Subtend
If the endpoints of a chord or arc lie on the sides of an inscribed angle, then the chord or arc is said to
subtend the angle.
Tangent
A line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.
Tangent Circles
Coplanar circles that intersect in one point.
Tangent Segment
A segment that is tangent to a circle at an endpoint.
Theorems and Corollaries
Angles Inside the
Circle Theorem
If two chords intersect inside a circle, then the measure of each angle is one -half the sum of the measures
of the arcs intercepted by the angle and its vertical angle.
Angles Outside the
Circle Theorem
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the
angle formed is one-half the difference of the measures of the intercepted arcs.
Arc Addition
Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
Circumscribed Angle
Theorem
The measure of a circumscribed angle is equal to 180 minus the measure o f the central angle that
intercepts the same arc.
Congruent Central
Angles Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding
central angles are congruent.
Congruent Circles
Theorem
Two circles are congruent circles if and only if they have the same radius.
Congruent
Corresponding Chords
Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding
chords are congruent.
Equidistant Chords
Theorem
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant
from the center.
Inscribed Angles of a
Circle Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Inscribed
Quadrilateral
Theorem
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
