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Chapter 6: Circles
6.1: Circle; Segments; Angles
6.2: Angle Measures in the Circle
6.3: Line and Segment Relationships
in the Circle
6.4: Constructions and Inequalities
in the circle
Definition:
A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle.
A
Radius :
A segment that joins the center of the circle to a point on the circle.
All radii of a circle are congruent.
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Congruent circles
Two or more circles that have congruent radii.
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Concentric circles
Two or more coplanar circles that have a common center.
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Chord
A line segment that joins two points of the circle.
Diameter
A chord that goes through the center of the circle.
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Central Angle
An angle whose vertex is the center of the circle and whose sides are radii of the circle.
Arc
Part of the circle.
The Intercepted Arc of an angle
Is determined by two points of intersection of the angle with the circle and all points of the arc in the interior
of the circle.
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Theorem 6.1.1
A radius that is perpendicular to a chord bisects the chord.
Given:
Prove:
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Proof:
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Postulate 16
In a circle, the degree measure of a central angle is equal to the degree measure of its intercepted arc.
The sum of the measures of the consecutive arcs that form a circle is 360°.
Definition
In a circle or congruent circles, congruent arcs are arcs with equal measures.
Postulate 17: Arc-Addition Postulate
If 𝐴𝐵 and 𝐵𝐶 intersect only at point B, then 𝑚𝐴𝐵 + 𝑚𝐵𝐶 = 𝑚𝐴𝐶 .
B
Definition:
An inscribed angle of a circle is an angle whose vertex is a point on the circle and whose sides are chords of the circle.
B
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Theorem 6.1.2
The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.
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Case 1
Case 2
Case 3
Theorem 6.1.3
In a circle (or congruent circles), congruent minor arcs have congruent central angles.
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Theorem 6.1.4
In a circle (or congruent circles), congruent central angles have congruent arcs.
Theorem 6.1.5
In a circle (or congruent circles), congruent chords have congruent minor (major) arcs.
Theorem 6.1.6
In a circle (or congruent circles), congruent arcs have congruent chords.
Theorem 6.1.7
Chords that are at the same distance from the center of the circle are congruent.
Theorem 6.1.8
Congruent chords are located at the same distance from the center of the circle.
Theorem 6.1.9
An angle inscribed in a semicircle is a right angle.
Theorem 6.1.10
If two inscribed angles intercept the same arc, then these angles are congruent.
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