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Chapter 11 Theorems
A set of theorems, postulates, and corrolaries from Chapter 11 in MATHEMATICS B on Circle Geometry (NY State).
29
Mathematics
10th Grade
02/13/2008

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Term

All radii of the same circle are _________.

Definition

All radii of the same circle are congruent

Term
A central angle is an angle whose vertex is ____________.
Definition
A central angle is an angle whose vertex is the center of the circle.
Term
What is a minor arc?
Definition

A minor arc is an arc that measures less that 180 degrees.

Term
What is a major arc?
Definition
A major arc is an arc whose measure is greater than 180 degrees.
Term
Arc Addition Postulate
Definition
mAB + mBC = mABC
Term

Theorem 52

Definition

In a circle or in congruent circles, congruent central angles intercept congruent arcs.

Term
Theorem 53
Definition
In a circle or in congruent cirlces, congruent arcs are intercepted by congruent central angles.
Term
Theorem 54
Definition
In a circle or in congruent circles, congruent central angles have congruent chords.
Term
Theorem 55
Definition
In a circle or in congruent circles, congruent arcs have congruent chords.
Term
Theorem 56
Definition
In a circle or in congruent circles, congruent chords have congruent central angles.
Term
Theorem 57
Definition

In a circle or in congruent cirlces, congruent chords have congruent arcs.

Term
Theorem 58
Definition
A diameter perpendicular to a chord bisects the chord and its arcs.
Term
Theorem 59
Definition
If two chords of a circle are congruent, they are equidistant from the center of the circle.
Term
Theorem 60
Definition

If two chords of a circle are equidistant from its center, they chords are congruent.

Term
Definition of an inscribed angle.
Definition

An inscribed angle equals half the measure of the intercepted arc

 

m ABC = 1/2 (mAC)

Term

Theorem 61

Definition
The measure of an inscribed angle of a circle is equal to one-half the measure of its intercepted arc.

Case 1

The center of the circle is a point on one ray of the inscribed angle


Case 2

The center of the circle is a point in the interior of the inscribed angle


Case 3 The center of the circle is a point in the exterior of the inscribed angle

Term
Corollary 61-1
Definition
An angle inscribed in a semicircle is a right angle.
Term
Corollary 61-2
Definition
Two inscribed angles of a circle that intercept the same arc are congruent.
Term

At any given point on a circle there is ___ and only ___ tangent to the circle

Definition
one and only one tangent to the circle
Term
Theorem 62
Definition
If a line is perpendicular to the radius at its point of intersection with the circle, then the line is tangent to the circle.
Term
Theorem 63
Definition
If a line is tangent to a circle, the line is perpendicular to the radius drawn to the point of tangency.
Term
Theorem 64
Definition
Tangent segments drawn to a circle from an external point are congruent.
Term
Corollary 64-1
Definition
If two tangents are drawn to a circle from an external point, the line determined by that point and the center of the circle bisects the angle formed by the tangents.
Term
Theorem 65
Definition

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.

 

m  SPR = 1/2 (mPR)

Term
Theorem 66
Definition

The measure of an angle formed by two chords intersecting within a circle is equal to one-half the sum (average) of the measures of the arcs intercepted by the angle and by its vertical angle.

 

m  AED = 1/2 (mAD+mBC)

Term
Theorem 67
Definition
The measure of an angle formed by Case 1, Case 2, or Case 3 (see below) is equal to half the difference of the measures of the intercepted arcs.

Case 1

An angle formed by a tangent and a secant


Case 2

An angle formed by two secants


Case 3

An angle formed by two tangents

Term
Theorem 68
Definition
If two chords intersect within a circle, the product of the measures of the segments of one chord equals the product of the measures of the segments of the other chord.
Term
Theorem 69
Definition
If a tangent and a secant are drawn to a circle from an external point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external segment.
Term
Theorem 70
Definition
If two secants intersect outside a circle, then the product of the measures of one secant segment and its external segment is equal to the product of the measures of the other secant segment and its external segment.
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