Bronx High School of Science
Ms. Abbott
Mathematics Department
M$4
Unit 6 (Part I): Circles
DEFINITIONS
 A circle is a set of points equidistant from a fixed point, the center.
 A radius is a segment from the center of the circle to any point on the circle.
 A chord is a segment connecting any two points on the circle.
 A diameter is a chord through the center of the circle.
 A tangent is a line that intersects the circle at one and only one point.
 A secant is a line that intersects the circle at two points.
 A central angle is an angle whose vertex is the center of the circle.
 Two circles are congruent if they have congruent radii.
 An arc is part of a circle.
 An arc is intercepted by an angle if the endpoints of the arc lie one on each ray of
the angle.
 A semicircle is half of a circle, intercepted by a diameter.
 A major arc is an arc greater than a semicircle, referred to with three letters:
ABC
 A minor arc is an arc less than a semicircle, referred to with two letters: AB
 The measure of an arc ( m AB ) is equal to the measure of the central angle
intercepting it.
 Congruent arcs are arcs of the same or congruent circles that are equal in
measure.
 An inscribed angle of a circle is an angle whose vertex is on the circle and whose
sides are chords of the circle.
POSTULATES
 Arc Addition Postulate: If AB and BC are arcs on the same circle, then
mAB  mBC  mABC
 There is one and only one line tangent to a circle at a given point on the circle.
THEOREMS and COROLLARIES
 In the same or congruent circles, all radii are congruent.
 In the same or congruent circles, congruent central angles intercept congruent
arcs.
 In the same or congruent circles, congruent arcs are intercepted by congruent
central angles.
 In the same or congruent circles, congruent chords have congruent arcs.
 In the same or congruent circles, congruent arcs have congruent chords.
 A radius perpendicular to a chord bisects the chord and its arcs.
 The perpendicular bisector of a chord passes through the center of the circle.
 If two chords of a circle are congruent, they are equidistant from the center of the
circle.
Bronx High School of Science
Ms. Abbott
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Mathematics Department
M$4
If two chords of a circle are equidistant from the center of a circle, they are
congruent.
The measure of an inscribed angle of a circle is equal to one half the measure of
its intercepted arc.
An angle inscribed in a semicircle is a right angle.
In the same or congruent circles, two inscribed angles are congruent if and only if
they intercept congruent arcs.
In a circle, parallel chords intercept congruent arcs between them.
If a line is tangent to a circle, the line is perpendicular to the radius at the point of
tangency.
If a line is perpendicular to a radius at its point of intersection with the circle, it is
tangent to the circle.
Tangent segments drawn to a circle from the same exterior point are congruent.
The measure of an angle formed by a tangent and a chord intersecting at a point of
tangency is equal to half the measure of the intercepted arc.
The measure of an angle formed by two chords intersecting within a circle is
equal to half the sum of the arcs intercepted by the angle and its vertical angle.
The measure of an angle formed by a tangent and a secant, two secants, or two
tangents intersecting outside a circle is equal to half the difference of the
intercepted arcs.