Name _____________________________________________________________________ Hour _________

Honors Geometry Lesson 10-1: Use Properties of Tangents

Learning Target: By the end of today’s lesson we will be able to successfully use properties of a tangent to a circle.

Vocabulary Definition Illustration

Circle A circle is the set of all points in a plane that are

_________________from a given point.

Center The center of a circle is the ____________ from

which all points of the circle are equidistant.

Radius

A _________________ from

the______________ of a circle to any point on

the circle is a radius.

Chord A chord is a ______________ whose

_________________ are on a circle.

Diameter A diameter is a ___________ that contains the

_____________ of the circle.

Secant

A secant is a ____________ that intersects a

circle in _______ points.

Tangent A tangent is a _________ in the plane of a circle

that intersects the circle in exactly _______ point.

Ex 1: Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.

a.)

b.)

c.)

Ex 2: Tell how many common tangents the circles have and draw them.

a.) b.) c.) d.) e.)

Ex 3: The diameter of L and M are 20 and 13 units, respectively, and QR = 4. Find each measure.

a) LQ b) RM

BC

EA

DE

Theorem 10.1 In a plane, a line is tangent to a circle if and only if the line is ____________________

to a ___________________ of the circle at its endpoint on the circle.

Ex 4: In each diagram, is a radius of circle R. Is tangent to circle R?

a.) TR = 26 b.) UT = 8

Ex 5: In the diagram, B is a point of tangency. Find the radius r of circle C.

Theorem 10.2:

Tangent segments from a common external point are____________________

Ex 6: is tangent to circle C at R and is tangent to circle C at S.

Find the value of x

Ex 7: is tangent to circle C at S and is tangent to circle C at T.

Find the value of x.

Ex 8: You are standing 36 feet from a circular swimming pool. The distance from you to a point of tangency on the pool

is 48 feet as shown. What is the radius of the swimming pool?

Ex 9: Consider a circle with radius 2. Two radii are drawn at right angles to each other. A tangent is drawn at each of the

points where the radii intersect the circle. How far from the center of the circle do the tangent lines intersect? Express

answer as a simplifies radical.

RS ST

QR QS

RS RT

Name _____________________________________________________________________ Hour _________

Honors Geometry Lesson 10-2: Find Arc Measures

Learning Target: By the end of today’s lesson we will be able to successfully use angle measures to find arc measures.

Vocabulary Definitions Illustrations

Central angle

A central angle of a circle is an angle

whose vertex is the ______________

of the circle.

Minor arc

Part of a circle measuring less than

__________.

Major arc

Part of a circle measuring between

______ and______.

Semicircle

A semicircle is an arc with endpoints

that are the endpoints of a

_______________. The measure of a

semicircle is _______.

Measure of a minor arc

The measure of a minor arc is the

measure of its _______________

_________________.

The expression is read as “the

________________ of arc AB”.

Measure of a major arc

The measure of a major arc is the

difference between ________ and the

measure of the related _____________

________.

Congruent circles

Two circles are congruent circles if

they have the same ___________.

Congruent arcs

Two arcs are congruent arcs if they

have the same ________________ and

they are arcs of the same circle or of

______________ circles.

Ex 1: Tell whether the given arcs are congruent. Explain why or why not.

a) b) c)

Postulate 23: Arc Addition Postulate

The measure of an arc formed by two ________________arcs is the ___________ of the

measures of the two arcs.

Ex 2: are diameters of A. Identify the following as a major arc, minor arc, or semicircle of the circle.

Then find its measure.

a) 𝑚𝑃𝑅�̂� b) 𝑚𝑆�̂� c) 𝑚𝑃𝑄�̂� d) 𝑚𝑅�̂�

e) 𝑚𝑈�̂� f) 𝑚𝑅�̂� g) 𝑚𝑈�̂� h) 𝑚𝑃𝑄�̂�

Ex 3: You join a new bank and divide your money several ways,

as shown in the circle graph at the right. Find the indicated arc measures

c. d.

***

Ex 4: Find m , x and y.

Ex 5: C is the center of the circle and mÐCQR = 4x°. Find the possible values of x.

Ex 6: Use the figure shown where = 45°, DN= 18, and . Find NG.

Ex 7: The device shown is a 10-second game timer. The top plunger button alternatively stops and starts the timer.

For game play, the timer is started at 10 (as shown) and moves counterclockwise. Players often start and stop the timer

several times before it reaches 0. Give all answers to the nearest tenth.

a) What is the measure of the arc traced out by the tip of the pointer as it moves

from one number to the next on the timer?

b) What is the measure of the arc traced out as the

pointer moves from the 10 to the 3?

c) A player starts the timer at the 10 and stops it after 7.4 seconds.

What is the measure of the arc generated?

d) A player stops the timer after 3.3 seconds, then after 1.7 seconds,

and again after 2.5 seconds. What is the sum of the measures of the arcs?

e) How much time does it take the pointer to trace out an arc of 80°?

QTandPR

OGNO

DH

DN

Name _____________________________________________________________________________ Hour _________

Honors Geometry Lesson 10-3: Apply Properties of Chords

Learning Target: By the end of today’s lesson we will be able to successfully use relationships of arcs and

chords in a circle.

Theorem 10.3 In the same circle, or in congruent circle, two minor arcs are ___________if and only

if their corresponding ______________ are congruent.

if and only if __________ _________.

Ex 1: Find the value of x.

a) b)

Ex 2:

Theorem 10.4:

If one chord is a ____________________ _________________of another chord,

then the first chord is a _____________________.

If is a perpendicular bisector of ,

then _____________is a diameter of the circle.

Ex 3: If x = 3, is the diameter of the circle? Explain.

Theorem 10.5

If a diameter of a circle is _______________ to a chord, then the diameter __________ the chord and its arc.

If is a diameter and ,

then and ________ ________.

Ex 4:

QS TR

BD

EG DFEG

HFHD

Ex 5: In the diagram below, is a diameter of O and is perpendicular to . If AB = 2 0 and CX = 2, what is the

radius of O?

Theorem 10.6:

In the same circle, or in congruent circles, two chords are _______________ if and only if

they are equidistant from the center.

if and only if ______ = _______.

Ex 6:

a) In the diagram of F, AB = CD = 12. Find EF. b) Use the diagram of P to find the length

of QR and TS.

CD AB

CDAB

Name _____________________________________________________________________ Hour _________

Honors Geometry Lesson 10-4: Use Inscribed Angles and Polygons

Learning Target: By the end of today’s lesson we will be able to successfully use inscribed angles of circles.

Vocabulary Definition Illustrations

Inscribed angle

An inscribed angle is an angle whose

____________ is on a circle and

whose sides contain ________ of the

circle.

Intercepted arc

The arc that lies in the interior of an

_______________ angle and has

_______________ on the angle is

called the intercepted arc of the angle.

Inscribed polygon

A polygon is an inscribed polygon if

all of its ______________ lie on a

circle.

Circumscribed circle

A circumscribed circle is a circle that

contains the vertices of an

______________ polygon.

Theorem 10.7: Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one half the measure of its_____________________ _______.

mADB = ½ • ___________

Ex 1: Find the indicated measure in P.

a.) mS b.) measure of arc RQ.

Ex 2: Find the indicated measure.

a.) mVT}

b.) Qm c.) x

Ex 3: Find the measure of arc HJ and HGJ.

What do you notice about HGJ and HFJ?

Theorem 10.8:

If two inscribed angles of a circle intercept the same arc, then the angles are _____________.

ADB ___________

Ex 4: Name two pairs of congruent angles in the figure and find their measures.

Theorem 10.9:

If a _________ triangle is inscribed in a circle, then the ____________is a diameter of the circle.

Conversely, if one side of an inscribed triangle is a ________________ of the circle, then the

triangle is a ___________ triangle and the angle opposite the diameter is the___________ angle.

mABC = 90 if and only if ______ is a diameter of the circle

Ex 5: Find all the angles in the triangle.

Theorem 10.10: A quadrilateral can be inscribed in a circle if and only if its opposite angles are_______________________.

D, E, F, and G lie on C if and only if mD + mF = mE + mG = ________.

Ex 6: Find the value of each variable.

a.) b.)

Name ____________________________________________________________________ Hour________

Honors Geometry Lesson 10-5: Apply Other Angle Relationships in Circles

Learning Target: By the end of today’s lesson we will be able to successfully find the measures of angles

inside or outside a circle.

Ex 1: Review: Use the figure below which shows a pentagon inscribed in O.

Assume and mABC = 132°.

a) mBAE

b) mAEB

c) mCOD

Ex 2: You are flying in an airplane about 5 miles above the ground.

What is the measure of arc BD, the part of Earth that you can see?

(Earth’s radius is about 4000 miles.)

Theorem 10.11:

If a ______________ and a _____________ intersect at a point on a circle, then the measure

of each angle formed is one half the measure of its intercepted arc.

m1 = ½ •______ m2 = ½ •______

Ex 3: Line m is tangent to the circle. Find the indicated measure.

a.) ml b.) m𝐷𝐹𝐸 ̂ and 𝑚𝐷�̂� c.) 2m

Theorem 10.12: Angles Inside a Circle Theorem: If two chords intersect _____________ a circle, then the measure of each angle is

one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

m1 = ½ • (m _______ + m _______ ) m2 = ½ •(m _______ + m _______ )

Ex 4: Find the value of x.

a.) b.)

AB @ BC @CD

Ex 5: Find mAOB in the figure below.

Theorem 10.13: Angles Outside of a Circle Theorem:

If a tangent and a secant, two tangents, or two secants intersect _______________ a circle, then the measure of the angle

formed is one half the difference of the measures of the intercepted arcs.

Ex 6: Find the indicated measure or the value of the variable or number.

a.) b.) c.)

Ex 7: In the figure below, is a diameter, m = 32°, m = 40°, and mEGD = 18°. Find mAFB.

AD CD AB

Name _____________________________________________________________________ Hour _________

Honors Geometry Lesson 10-6: Find Segment Lengths in Circles

Learning Target: By the end of today’s lesson we will be able to successfully find segment lengths in circles.

Vocabulary Definition Illustration

Segments of a chord

When two chords intersect in the

________________ of a circle, each

chord is divided into______ segments

called segments of the chord.

Secant segment

A secant segment is a _____________

that contains a chord of a circle, and

has exactly one

_____________outside the circle.

External segment

An external segment is the part of a

secant segment that is

__________________ the circle.

Theorem 10.14: Segments of Chords Theorem

If two chords intersect in the ________________ of a 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.

EA • ________ = EC • ________

OR…… _____ • _____ = _____• _____

Ex 1: Find x, ML and JK.

Theorem 10.15: Segments of Secants Theorem If two secant segments share the same endpoint _______________ a circle, then the product of the lengths of one secant

segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

(Whole secant segment • outside segment) = (Whole secant segment • outside segment)

EA • ____ = EC • ____

Ex 2: Find the value of x and RT.

Ex 3: Find EB and ED

Theorem 10.16: Segments of Secants and Tangents Theorem:

If a secant segment and a tangent segment share an ____________ ____________a circle,

then the product of the lengths of the secant segment and its external segment

equals the square of the length of the tangent segment.

(Tangent segment) 2 = (Whole secant segment • outside segment)

EA2 = ______ • _______

Ex 4: Find AL.

Ex 5: You are standing at point C, 45 feet from the Point State Park fountain in Pittsburgh.

The distance from you to a point of tangency on the fountain is 105 feet.

a) Find the distance CA between you and your friend at point A.

b) Find the radius of the fountain.