CirclesChapter 6

Sir Migo Mendoza

Central AnglesLesson 6.1

Sir Migo Mendoza

Central Angles

Definition 5.1 Arc

•An arc is a part of a

circle.

Types of Arc

•Minor Arc

•Major Arc

•Semicircle

Definition 5.2 Central

Angle

•A central angle of a circle is an angle

whose vertex is the center of the

circle.

Definition 5.2 Central

Angle

Definition 5.3 Minor Arc

•Minor arc consists of points M and I and all points of ⊙O that are

in the interior of central angle MOI.

Definition 5.3 Minor Arc

Definition 5.4 Major Arc

•Major arc consists of points M and I and

all points of ⊙O that are in the

exterior of central angle MOI.

Definition 5.4 Major Arc

Definition 5.5 Semicircle

•Semicircle consists of endpoints D and M of diameter DM and all points ⊙O that lie on

one side of DM.

Definition 5.5 Semicircle

Note:•The degree

measure of an arc is defined in

terms of its central angle.

Definition 5.6 Degree Measure of Minor

•The degree measure of minor is equal to

the degree measure of central angle MOI.

Definition 5.6 Degree Measure of Minor

Definition 5.7 Degree Measure

of Major

•The degree measure of major is equal to

360 minus the degree measure of central angle MOI.

Definition 5.7 Degree Measure

of Major

Definition 5.8 Degree Measure of Semicircle

•The degree measure of

semicircle DIM is equal to 180.

Definition 5.8 Degree Measure of Semicircle

Definition 5.9 Congruent

Circles

•Congruent Circles are

circles with the same radius.

Definition 5.10 Congruent Arc

•Congruent Arcs are arcs with the same measures.

Postulate 5.1 The Central Angle-

Intercepted Arc Postulate (CA-IA

Postulate)

•The measure of a central angle of a circle is equal to

the measure of its intercepted arc.

Example

Postulate 5.2 The Arc Addition

Postulate

Postulate 5.3

•A diameter divides a circle into two

semicircles.

Definition 5.11 Arc Length

• The measure of the central angle can also be used to determine the arc

length. The arc length (or length of an arc) is different from the degree

measure of an arc. That is, if a circle is made up of string, the length of the arc

is the linear distance of the piece of string representing the arc. The length of the arc is a part of the circumference and proportional to the measure of the

central angle when compared to the entire circle.

Example:

Let’s Practice:

Let’s Practice:

Let’s Practice:

Inscribed AngleLesson 6.2

Sir Migo Mendoza

Definition 5.12 Inscribed Angle

•An inscribed angle in a circle is an angle whose

vertex is on the circle and whose sides contain chords of the circle.

Example:

• Considering the definition, which

among these three is/are inscribed

angle?

Inscribed Angle?

Theorem 5.1 The Inscribed

Angle Theorem

•The measure of an inscribed angle is

one-half the measure of its

intercepted arc.

Proof:

Example:

Theorem 5.2 The Semicircle

Theorem

•An angle inscribed in a semicircle is a right angle.

Theorem 5.3 Inscribed Angles

in the Same Arc Theorem

•Two or more angles inscribed in the same arc are congruent.

Example:

Prove This:

Direction: Use the given figures

to find the value of x.

TangentsLesson 6.3

Sir Migo Mendoza

Definition 5.13 Tangent to a

Circle

•A line in the plane of the circle that

intersects the circle at exactly one point

is called tangent line.

Definition 5.14 Point of

Tangency

•It is the point of intersection between a tangent line and a

circle.

Note:• A circle separates a

plane into three parts:

1. the interior;

2. the exterior; and

3. the circle itself.

Theorem 5.4 The Tangent-Line

Theorem

•If a line is tangent to a circle, then it is

perpendicular to the radius at its outer

endpoint.

Theorem 5.4 The Tangent-Line

Theorem

Theorem 5.5 The Converse of

the Tangent-Line Theorem

• In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the

circle, then the line is a tangent to the circle.

Note:

•Theorem 5.5 can be used to identify

tangents to a circle.

Example:

Theorem 5.6 The Tangent-

Segment Theorem• If two tangent segments are

drawn to a circle from an external point, then:1. the two tangent segments

are congruent, and2. the angles between the

tangent segments and the line joining the external point to the center of the circle are congruent.

Proof:

Definition 5.14 Common

Tangent

• A line or a segment that is tangent to two

circles in the same plane is called a

common tangent of the two circles.

Types of Common Tangents

1. Common External Tangent

2. Common Interior Tangents

Common External Tangent

•Common external tangents do not intersect

the segment whose endpoints are the centers

of the circles.

Common External Tangent

Common Interior Tangents

•Common interior tangents intersect

the segments whose endpoints are the

centers of the circles.

Common Interior Tangent

Theorem 5.7 The Tangent

Circles Theorem• If two circles are tangent internally or externally, then their line of centers

pass through the point of contact.

Internally Tangent Circles

Externally Tangent Circles

Proof:

Definition 5.15 Tangent Circles

•These are two circles whose intersection is

exactly one point.

Definition 5.16 Line of

Centers

•It is the segment joining the

centers of two circles.

Definition 5.17 Common Tangent

•It is a line which is tangent to two

circles.

Definition 5.18 Common

Internal Tangent

•It is a common tangent which intersects the

line of centers.

Definition 5.19 Common

External Tangent

•It is a common tangent which

does not intersect the line of centers.

Definition 5.20 Internally

Tangent Circles

•These are tangent circles whose

common tangent does not intersect the line of centers.

Definition 5.21 Externally

Tangent Circles

•These are tangent circles whose

common tangent intersects the line of

centers.

Prove This:

Let’s Practice:

Chords and ArcsLesson 6.4

Sir Migo Mendoza

Theorem 5.8

•The perpendicular from the center of the circle to any chord bisects the

chord.

Proof:

Theorem 5.9• The line joining the center of the circle to the midpoint of any chord which is not a

diameter is perpendicular to the

chord.

Proof:

Theorem 5.10

•The perpendicular bisector of a

chord of a circle passes through

the center of the circle.

Proof:

Theorem 5.11•The perpendicular bisector of a chord of a circle bisects the central angle subtended by the

chord.

Proof:

Theorem 5.12• The bisector of a

central angle subtended by the

chord is the perpendicular bisector

of the chord.

Proof:

Theorem 5.13• In the same circle or in

congruent circles, chords are congruent if and only if their

distances from the center(s) of the circle(s) are equal.

• Note: Congruent circles are those whose radii are

congruent.

Proof:

Theorem 5.14• In a circle or in

congruent circles, two minor arcs are

congruent if and only if their corresponding chords are congruent.

Proof:

Direction: Find x in each figure.

Angles Formed by Secants,

Tangents, and ChordsLesson 6.5

Sir Migo Mendoza

Introduction:•There are four ways

for the two intersecting lines to

intersect a circle. These are:

(1) Inscribed Angle

(2) an angle formed by a tangent and a

secant;

(3) an angle formed by two secants

intersecting in the interior of the circle; and

(4) an angle formed by two secants

intersecting in the exterior of the circle.

Theorem 5.15 The Intersecting

Secants-Exterior Theorem

• The measure of an angle formed by two secants that intersect

in the exterior of a circle is one-half the

difference of its intercepted arcs.

Proof:

Example:• The angle between two secants intersecting in the exterior of the circle is 55°. If one of the intercepted

arcs measures 150°, what is the degree measure of the

other arc?

Theorem 5.16

• The measure of an angle formed by a

tangent and a secant drawn at the point of contact is one-half the

measure of its intercepted arc.

Proof:

Example:

Theorem 5.17 The Intersecting

Secants-Interior Theorem

• The measure of an angle formed by two secants

intersecting in the interior of the circle is equal to one-half the

sum of the measures of its intercepted arcs.

Example:

Example:

Let’s Practice:

Let’s Practice:

The Power TheoremsLesson 6.6

Sir Migo Mendoza

Theorem 5.18 The Intersecting

Segments of Chords Power Theorem

• If two chords intersect in the interior of the circle, then the product of the

lengths of the segments of one chord is equal to the product of the lengths of the segments of the other

chord

Example:

Theorem 5.19 The Segments of

Secants Power Theorem

• If two secants intersect in the exterior of the circle, the

product of the length of one secant segment and the

length of its external part is equal to the product of the length of the other secant

segment and the length of its external part.

Example:

Theorem 5.20 The Tangent Secant

Segments Power Theorem

• If a tangent segment and a secant intersect in the exterior of a circle, then the square of

the length of the tangent segment is equal to the

product of the lengths of the secant segment and its

external part.

Example:

Let’s Practice: Find the value of x.