Honors Geometry Chapter 6 Tentative Syllabus - Team … · Red Day Red Date Blue Day Blue Date Topic ... p.320, #1–13, 15–19 Tue Nov 24 Mon ... Study for Final Exam Tue Dec 15

Created by W.L. Bass and used with permission, p.1

Honors Geometry – Chapter 6 Tentative Syllabus

YOU ARE EXPECTED TO CHECK YOUR ANSWERS BEFORE COMING TO CLASS.

Red

Day

Red

Date

Blue

Day

Blue

Date

Topic

Homework

(due next class period)

Fri

Nov 20

Mon

Nov 23

6.1 Tangent Properties

6.2 Chord Properties

In-class: 6.2 Wkbk p.40, #1–9

6.1 Wkbk: p.39, #1– 6

p.320, #1–13, 15–19

Tue

Nov 24

Mon

Nov 30

6.3 Arcs and Angles

6.4 Other Angles

In-class: 6.3 Wkbk… p.41

Circle Angle Chase #1 (Notes p.20)

p.327, #1–16, 22–24, 26

Circle Angle Chase #2 &

#3

(Notes p.21 and 22)

Tue

Dec 1

Wed

Dec 2

6.5 Circumference & Diameter

6.6 Around the World

In-Class: 6.5 Wkbk… p.43

p.337, #1–13

p.342, #2, 4, 6

Review for Final Exam #1 ( DUE on exam day)

Thurs

Dec 3

Fri

Dec 4

6.7 Arc Length

In-class: 6.7 Wkbk… p.45

p.351… 1–9, 17

Circle Angle Chase #4 (Notes p.23)

Mon

Dec 7

Tue

Dec 8

13.5 Indirect Proof Notes Review: Chp 6 Practice Test

p.359… 4–20, 25–28, 31, 32, 35, 38, 46, 47, 52, 53,

62, 63

Wed

Dec 9

Thurs

Dec 10

Chapter 6 Test

Indirect Proof Practice

Final Exam Review #2 (Due day of Final Exam)

Fri

Dec 11

Mon

Dec 14

Final Exam Review

Study for Final Exam

Tue

Dec 15

Wed

Dec 16

Enjoy Winter Break!


Section Indiana Standard Learning Target

6.1

6.2

GCl1 Define, identify, and use relationships

among the following: radius, diameter, arc,

measure of an arc, chord, secant, tangent, and congruent concentric circles.

GCl3 Identify and describe relationships among

inscribed angles, radii, and chords, including the following: the relationship exists between

central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles;

and the radius of a circle is perpendicular to a

tangent where the radius intersects the circle. GCl6 Construct a tangent line to a circle through

a point on the circle, and construct a tangent line

from a point outside a given circle to the circle;

justify the process used for each construction.

Review and use basic properties of a circle

and circle vocabulary.

Discover and use properties of tangents of circles.

Construct a tangent line (Investigation 2)

Discover and use properties of chords in a circle.

(p.322 #23)

6.3 GCl1 Define, identify, and use relationships among the following: radius, diameter, arc,

measure of an arc, chord, secant, tangent, and

congruent concentric circles. GCl4 Solve real-world and other mathematical

problems that involve finding measures of

circumference, areas of circles and sectors, and arc lengths and related angles (central, inscribed,

and intersections of secants and tangents).

Discover and use relationships between an inscribed angle of a circle and its

intercepted arc.

Construct a cyclic quadrilateral. Know and use that the opposite angles of a

cyclic quadrilateral are supplementary.

(Investigation 4)

6.4 GLP4 Develop geometric proofs, including direct

proofs, indirect proofs, proofs by contradiction and proofs involving coordinate geometry, using

two-column, paragraphs, and flow charts

formats.

Prove and use circle conjectures.

6.5 GCl1 Define, identify, and use relationships among the following: radius, diameter, arc,

measure of an arc, chord, secant, tangent, and

congruent concentric circles.

Calculate pi, the ratio of the circumference of a circle to its diameter.

6.6 GCl4 Solve real-world and other mathematical

problems that involve finding measures of circumference, areas of circles and sectors, and

arc lengths and related angles (central, inscribed,

and intersections of secants and tangents). GQP5 Deduce formulas relating lengths and

sides, perimeters, and areas of regular polygons.

Understand how limiting cases of such formulas lead to expressions for the circumference and the

area of a circle.

Apply circle properties and the

circumference of a circle to solve problems. (p.343 #9)

6.7 GCl4, GCl5 Construct a circle that passes

through three given points not on a line and justify the process used.

Discover and use a formula for finding the

length of an arc of a circle. (p.354 #24)


BA

C

6.1 and 6.2 Notes

Tangent and Chord Properties

Big ideas about TANGENTS!

1. A tangent and a radius are perpendicular

to each other at the point of tangency.

Label the picture based on the property.

2. Tangent segments from the same point

outside a circle are congruent.


3. Recall: The measure of an arc is equal

to the measure of its central angle.

If 115m ABC then

𝑚 𝐶�̂� = ______________

4. Recall: All the radii (plural of radius)

are congruent to each other in a circle.


Apply the tangent theorems… Assume that lines that appear tangent are tangent.

1. 2. 3.

4. 5. 6.

x 255 x

33

x 65

x E 106

x 83 x

E

15


7. One more application

a.) Write an equation for the tangent line.

Find slope of the radius

Find slope of the tangent line (Hint: the radius and tangent are perpendicular)

Use the point-slope formula to write the equation for the tangent line.

1 1y y m x x

Rewrite the equation in slope-intercept form

b.) YOU TRY… Write an equation for the tangent line.

A little more vocabulary… Internally and Externally Tangent

Circles are considered internally or externally tangent when…

they are _________________ to the same _____________ at the same ______________.

Created by W.L. Bass and used with permission, p.5 H

D

G

FE

Big ideas about CHORDS

1. VOCABULARY – Central angle v. Inscribed Angle.

Central Angle – An angle that has its vertex on the ______________ of the circle

Inscribed Angle – An angle that has its vertex __________ the circle

AND the sides of the angle are __________________ of the circle.

Use the definitions above and the diagram at the below name the following…

Central Angles: __________________________________ Inscribed Angles: __________________________________

2. If two chords in a circle are congruent,

then their central angles are congruent.

Use your compass to show DE GH

Draw in radii FD , FE , FG , FH

DFE GFH by _____________

Therefore, DFE GFH

by _____________

3. If two chords in a circle are

congruent,

then their arcs are congruent.

From the “proof” above in Big Idea #2,

we saw that DFE GFH

Recall: The measure of an arc is ____________________ to the measure

of its central angle.

Therefore, if the central angles are

congruent, then 𝐷𝐸̅̅ ̅̅ ≅ 𝐺𝐻̅̅ ̅̅

Label the picture

based on the Big Ideas 2 & 3.

H

O

U

N

D

C

O

R

K

M

A

T

H


H

D

G

FE

N

O

PQ

4. The perpendicular from the center of a

circle to a chord bisects the chord.

Construct a perpendicular from F

to DE and label the intersection M.

Use your compass to show DM ME

Construct a perpendicular from F

to GH and label the intersection P

Use your compass to show GP PH

5. If two chords in a circle are congruent,

then they are equidistant from the

center.

Recall: Distance from a point to

line/segment must be measured on a _______ line.

Since we’ve already constructed the

perpendicular in Big Idea #4…

Use your compass to show FM FP .

Label the picture

based on Big Ideas 4 & 5.

6. The perpendicular bisector of a chord is also the diameter of the circle.

Construct the perpendicular bisector of NO … Name the bisector m.

Construct the perpendicular bisector of PQ … Name the bisector p.

Label the intersection of m and p point F.

Summary:

Both ______ and ______ contain diameters of the circle.

Point ______ is the _____________of the circle!


I

N

OP

6.3 Notes

Arcs and Angles

Review/Warm-up… Write the equation for the tangent line shown below.

6.3 BIG IDEAS!

1. VOCABULARY – Central angle v. Inscribed Angle.

Central Angle – An angle that has its vertex on the ______________ of the circle

Inscribed Angle – An angle that has its vertex __________ the circle

AND the sides of the angle are __________________ of the circle.

2. The measure of an inscribed angle is one-half the measure of its intercepted arc.

Use a piece of patty paper to trace 2 m N

Draw central angle OIP

Compare 2 m N to m I

Recall: The measure of an arc is ____________________ to the measure of its central angle.

Summary:

_____________ _____________m I

1

______________2

m I

2 ______________m N

Measure of Inscribed Angle = ___________________

Measure of Intercepted Arc = ___________________


I

N

OP

R

U

TS

W

Y Z

W

X

3. Inscribed angles that intercept the

same arc (or congruent arcs) are congruent.

The intercepted arc for O is _________

The intercepted arc for P is _________

Use a piece of patty paper to show O P

What about N and R ?

4. Parallel lines (segments) intercept

congruent arcs.

If / /ST WU , then 𝑆�̂� ≅ _____________

Why? Draw transversal SU .

TSU WUS because…

______________________________

Look again at Big Idea #3...

Congruent Arcs Congruent angles

5. Angles inscribed in a semicircle are right angles.

Draw inscribed YWZ

What’s the measure of its intercepted arc?_________

Therefore __________m W

Draw inscribed ZXY

What’s the measure of its intercepted arc?_________

Therefore __________m Z

What kind of triangles are

YWZ and ZXY ? _________________________.

Are they congruent to each other? _____________.

Are they isosceles? _____________.


I

N

O

P

R

6. The opposite angles in a cyclic quadrilateral are supplementary.

Definition: Cyclic quadrilateral

____________________________________________

Examine m N m R … What do you

notice?__________________________

Therefore…

N and ______________ are supplementary

Use a piece of patty paper to trace m O m P …

What do you notice?__________________________

Therefore…

O and ______________ are supplementary

Apply what you have learned… Solve for the variable in each diagram.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. Hint: Write the formula first. 11.

x˚

y˚

x˚

105˚

(2x + 11)˚

(4x - 3)˚

A

B

C

D

48˚ C

B

A

x˚ B

A

C

87˚ x˚ C A

B

28˚

x˚

37˚

B

A

C x˚

208˚

(5x + 4)˚

142˚

C B

A D

x˚

A

B

C

138˚

x˚

x˚

y˚

110˚


Ch 6 Exploration

Other Angle Measures

Match the conjecture to the appropriate diagram below then write an equation to find in marked angle.

1.___________ Intersecting Secants Conjecture: The measure of an angle formed by 2 secants that intersect

________ the circle is ___________________

2.___________ Intersecting Chords Conjecture: The measure of angle formed by 2 intersecting

______________ (or 2 secants intersecting inside) is ___________________

3.___________ Tangent-Secant Conjecture: The measure of an angle formed by an intersecting tangent and a

secant to a circle is ___________________

4.___________ Intersecting Tangents Conjecture: The measure of angle formed by 2 intersecting

______________ to a circle is ___________________

5.___________ Tangent Chord Conjecture: The measure of an angle formed by the intersection of a tangent

and a chord at the point of tangency is ___________________

A. B. C.

D. E.

T

B

E

1

A

T

B

E

1 A

T

S

B

E

1

A

A

B

C

D

1

Z

P

A

E

1


Apply what you know…

1. Finding Angle and Arc Measures

a.) Given: 𝑚𝐵�̂� = 54°, mark this in the picture.

b.) 𝑚 𝐶𝐸�̂�= ____________

c.) Find the intercepted arc for ABC and m ABC = ____________

d.) Find the intercepted arc for CBD and m CBD = ____________

e.) What do you notice about ABC and CBD ?

2. Finding the Measure of an Angle Formed by Two Chords

a.) Solve for x .

b.) Solve for x .

A B

D

C

E

E is not the center!

A

B

E

C

D

24 100 x

E is not the center!

center!

A

B

E

C

D

30 75 x


3. Finding angles and arcs when intersection is OUTSIDE the circle.

a..) Solve for x .

Mark the intercepted arcs

b.) Solve for x .


c.) Solve for x .


𝑚𝐴�̂�= ____________,

d.) Solve for x .


𝑚𝐶�̂� = ____________,

e.) Solve for x .


𝑚𝐴𝐶�̂�= ____________,

f.) Solve for x .


𝑚𝐴𝐶�̂�= ____________,

6.5 and 6.6 Notes

A

B

C

D

110 x 49

x C

B

E A

56

x 125

A

B

C

x 52

A

B

C

A

B

C

D

114 x 50

x

C

B

E A

36


Circumference/Diameter, Around the

World

6.5 – Circumference Conjecture

If C is the circumference and d is the diameter of a circle, then there is a number such that

_________C .

If r is the radius then _________d and _________C

1. If 7C m , find the radius.

2. If 8.5d m , find the circumference

(no decimals).

3. If 36C m , find the radius.

4. If 6.25d m , find the circumference

(no decimals)

6.6 Practice…

5. A satellite in a nearly circular orbit is 2000 km above the Earth’s surface. The radius of the Earth is approximately 6400 km. If the satellite completes its orbit in 12 hours, calculate the speed of the

satellite in km per hour.

6. The diameter of a car tire is approximately 60 cm (0.6 m). The warranty is good for 70,000 km. about

how many revolutions will the tire make before the warranty is up? More than a million? More than a billion? (1 km = 1000m)

7. Goldi’s Pizza Palace is known throughout the city. The small Baby Bear pizza has a 6-inch radius and

sells for $9.75. The savory medium Mama Bear pizza sells for $12.00 and has an 8 inch radius. The

large Papa Bear pizza is a hefty 20 inches in diameter and sells for $16.50. The edge is stuffed with cheese and it’s the best part of the pizza. What size has the most pizza edge per dollar? What is the

circumference of this pizza?

6.7 Notes

Arc Length


U

G

H50

9

6.7 – Finding Arc Length v. Arc Measure

The ___________________ of an arc is equal to…

Definition…

Proportion…

Find the indicated measures.

1.

𝑚𝐺�̂� = _________ 𝑚𝐻𝑈�̂� = __________ length of 𝐻𝑈�̂�=___________


R

A

T

70

P

1160

Y

U

P

2.

𝑚𝑌�̂� = ___________ 𝑚𝑌𝑈�̂� = __________ length of 𝑌�̂� =______________

3. WORKING BACKWARDS…

Given: 𝐴�̂� = 16𝜋

𝑚𝐴�̂� =___________𝑚𝐴𝑅�̂�=_____________Diameter = ___________


13.5 Notes

Indirect Proof

General steps for an indirect proof…

A.

B.

1. Given: 1 is not to 2

Prove: p is not // n

Statement Reason

a.) ____________________________ a.) ____________________________

b.) ____________________________ b.) ____________________________

c.) ____________________________ c.) ____________________________

d.) ____________________________ d.) ____________________________

2. Given: 1 67m

Prove: BX is not to AC

Statement Reason

a.) ____________________________ a.) ____________________________

b.) ____________________________ b.) ____________________________

c.) ____________________________ c.) ____________________________

d.) ____________________________ d.) ____________________________

e.) ____________________________ e.) ____________________________

2 1

p

n

1

X

B

A C


3. Given: 1 is not to 2

Prove: ABC is not isosceles with vertex angle B

Statement Reason

a.) ____________________________ a.) ____________________________

b.) ____________________________ b.) ____________________________

c.) ____________________________ c.) ____________________________

d.) ____________________________ d.) ____________________________

4. Given: OJ OK

JE is not to KE

Prove: OE does not bisect JOK

Statement Reason

a.) ____________________________ a.) ____________________________

b.) ____________________________ b.) ____________________________

c.) ____________________________ c.) ____________________________

d.) ____________________________ d.) ____________________________

e.) __________OJE e.) ____________________________

f.) ____________________________ f.) ____________________________

g.) ____________________________ g.) ____________________________

h.) ____________________________ h.) ____________________________

E

J

1 2

O

K

1 A

2 C

B


31

96E

BC

D

A

Circle Angle Chase #1

Given: EC is a diameter

For each problem, determine the measure of the arc or angle. Be careful to look where your vertex is.

One mistake can lead to many! Write your answers on the blanks.

Write your answers on the blanks provided.

1. m EBA _________

2. mEA _________

3. mAC _________

4. mED _________

5. mDC _________

6. m E _________

7. m D _________

8. m ECA _________

9. m A _______


# 11

# 5

# 12# 10

# 7

# 6

# 9

# 8

# 4

# 3

# 13

# 14

# 2

# 1

70

85


Given: AB is a diameter


One mistake can lead to many!


1. _________

2. _________

3. _________

4. _________

5. _________

6. _________

7. _________

8. _________

9. _________

10. _________

11. _________

12. _________

13. _________

14. _________

A

B


# 6

# 5

# 13 # 14

# 12# 15

# 10# 11

# 9

# 7

# 1

# 8

# 2

# 4

# 3

37

40

E

A

C

D


Given: //AC DE




1. _________

2. _________

3. _________

4. _________

5. _________

6. _________

7. _________

8. _________

9. _________

10. _________

11. _________

12. _________

13. _________

14. _________

15. _________






1. _________

2. _________

3. _________

4. _________

5. _________

6. _________

7. _________

8. _________

9. _________

10. _________

11. _________

12. _________

13. _________

14. _________

42˚

44˚

18˚

#1

#2

#3

#4

#5

#7

#6

#9

#10

#8

#11

#12

#13

#14


CHAPTER 6 CIRCLE SUMMARY

1.

4.

7.

10.

2.

5.

8.

11.

3.

6.

9.

Other things to remember…

Congruent chords

_____________________

Congruent arcs

_____________________

How to find the center of

a circle with only 3 points

How to write the

equation of a tangent line.

Circumference Formula

Arc Length

Documents

Honors Geometry Chapter 6 Tentative Syllabus - Team … · Red Day Red Date Blue Day Blue Date Topic ... p.320, #1–13, 15–19 Tue Nov 24 Mon ... Study for Final Exam Tue Dec 15