Download definitions and theorems 6 - The Bronx High School of Science

Bronx High School of Science
M$4
Mathematics Department
Ms. Abbott
Unit 6: Part I
DEFINITIONS:
 A circle is a set of points in a plane such that the points are equidistant froma
fixed point called the center of the circle.
 A radius of a circle is a line segment from the center of the circle to any point of
the circle.
 A central angle of a circle is n angle whose vertex is the center of the circle.
 An arc is intercepted by an angle if each endpoint of the arc is on a different ray
of the angle and the other points of the arc are in the interior of the angle.
 If an arc is less than a semicircle, it is a minor arc.
 If an arc is greater than a semicircle, it is a major arc.
 A semicircle is half of a circle.
 The degree measure of an arc is equal to the measure of the central angle that
intercepts the arc.
 Congruent circles are circles with congruent radii.
 Congruent arcs are the arcs of the same are 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 contain chords of the circle.
 A tangent to a circle is a line in the plane of the circle that intersects the circle in
exactly one point.
 A secant of a circle is a line that intersects the circle at two points.
 A tangent segment is a segment of a tangent line, one of whose endpoints is the
point of tangency.
POSTULATES:
 Arc Addition Postulate: If AB and BC are arcs of the same circle having a
common endpoint and no other points in common, then mAB  mBC  mABC .
 At a given point on a circle, there is one and only one tangent to the circle.
THEOREMS:
 All radii of the same circle are congruent.
 In a circle or in congruent circles, congruent central angles intercept congruent
arcs.
 In a circle or in congruent circles, congruent arcs are intercepted by congruent
central angles.
 In a circle or in congruent circles, congruent arcs have congruent chords.
 In the same or congruent circles, congruent chords have congruent arcs.
 A diameter perpendicular to a chord bisects the chord and its arcs.
 If two chords are congruent, they are equidistant from the center of the circle.
 The perpendicular bisector of a chord passes through the center of the circle.
Bronx High School of Science
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Mathematics Department
Ms. Abbott
If two chords are equidistant from the center of a circle, then the chords are
congruent.
The measure of an inscribed angle of a circle is equal to one-half the measure of
its intercepted arc.
If a line is perpendicular to a radius at its point of intersection with the circle, then
line is tangent to the circle.
If a line is tangent to a circle, the line is perpendicular to the radius drawn to the
point of tangency.
An angle inscribed in a semicircle is a right angle.
The inscribed angles of a circle that intercept the same arc are congruent.
Tangent segments drawn to a circle from an external point are congruent.
The measure of an angle formed by a tangent and a chord intersecting at the point
of tangency is equal to one-half the measure of the intercepted arc.
The measure of an angle formed by two chords intersecting within a circle is
equal to one-half the sum of the measures of the arcs intercepted by the angle and
its vertical angle.
The measure of an angle formed by a tangent and a secant, or two secants, or two
tangents intersecting outside a circle is equal to one-half the difference of the
intercepted arcs.