Unit 11-Aim 1: How can we find the measure of central angles?
Motivation: Student demonstration-use a rolling chair
1. “Johnny” is driving a car. The road is icy. “Johnny” turns the wheel, loses control and spins around one time. What is the name given to describe this rotation?
Answer: A “360”.
2. What do we know about circles?
Answer: There are 360

degrees in a whole circle
3. An arc of a circle is part of the circle between two points on the circle. In the diagram below of circle
O , radii AO and CO are drawn such that : :

3: 4:5 . Find the measure of each arc.
Step I: Equation
3 x

4 x

5 x

360
12 x
 x

30
360
Step II: Substitution
ANSWER: mBC

 

90
   
120 mAB
 check:
  
150
90

120

150

360

4. (part of) JAN 2008:36 In the diagram below of circle O , chords DF , DE , FG , and EG are drawn such that : : :

5 : 2 :1: 7 . Find the measure of each arc.
Step I: Equation
5 x

2 x

1 x

7 x
15 x
 x

24
360

360
Step II: Substitution
ANSWER:
 mFE
 mEG


  
120
  
48
  
24
  
168
5. A central angle of a circle is an angle whose vertex (point) is the center of the circle. a) Find the relationship between the central angles and their intercepted arcs (arcs between the angles). b) Find x . m BC
 m

BOC
90

90

 m

AOC
180

180
 m FC m


FOC
60
 x

Fill in:
The degree measure of an arc measure of an arc is equal to the measure of the central angle that intercepts the arc.
6. In the accompanying diagram of circle O , chord AB is parallel to chord CD . If , mAC

100 , find a) b) mBD m

BOD c) Fill in:
The measures of the arcs intercepted between parallel chords of a circle are equal.
Remember: “hamburger”

Note: Numbers may be equal, but geometric figures may be congruent (same size and same shape).
7. In the accompanying diagram, CD is parallel to diameter AB . If m AC

25 , what is

?
8.
AUG 2009:4 In the diagram of circle O below, chord CD is parallel to diameter
and mAC

30 .
What is mCD ?
1) 150 2) 120 3) 100
Step I: Theorem
In a circle, parallel lines intersect congruent arcs. remember: “hamburger”
4) 60
Step II: Definition
If the endpoints of an arc are the endpoints of a diameter, then the arc is a semi-circle.
Step III: Theorem
The measure of a semi-circle is 180

.

30 x

 x

60
 x

120
30
180

180
ANSWER: 2) 120

9. JUNE 2010:1 In the diagram below of circle O , chord AB chord CD , and chord CD chord EF .
Which statement must be true?
1) CE

DF
Step I: Diagram
2) AC

DF 3) AC

CE 4) EF

CD
Step II: Theorem
Parallel chords intersect congruent arcs.
ANSWER: 1) CE

DF
PI GG51 CIRCLES 70 Lesson #11
AIM: What are the parts of a circle?
Students will be able to
1. define circle, radius, diameter, center, chord, secant, tangent, central angle, arc, semicircle, minor arc, major arc, congruent arcs, and congruent circles
2. discover, state, and apply the postulates: a. In the same or congruent circles all radii are congruent b. The degree measure of a central angle of a circle is equal to the degree measure of its intercepted arc.
3. explain the difference between arc degrees and arc length
4. solve numerical and algebraic problems involving diameters and radii, major and minor arcs, and central angles
5. apply the above definitions and postulates to formal and informal proofs
Writing Exercise:
1. Which do you think is a better illustration of a circle: a round pizza or a bicycle tire? Explain the reasons for your choice.
2. Ellen wondered why a circle with a radius of 5 inches and a bigger circle with a radius of 8 inches both had 360 degrees. She asked Gary, “Shouldn’t the bigger circle have more degrees?” How should Gary answer her question?
In the accompanying diagram,

70,

70, and BOD is the diameter of circle O . Write an explanation or a proof that shows 
RBD and 
YDB are congruent

PI GG51 72 Lesson #13
AIM: How do we prove arcs congruent?
Students will be able to
1. state and apply the following: a. The degree measure of a central angle of a circle is equal to the degree measure of its intercepted arc. b. In the same or in congruent circles, if two central angles are congruent, then the arcs they intercept are congruent. c. In the same or in congruent circles, if two arcs are congruent, then their central angles are congruent. d. In the same or in congruent circles, if two chords are congruent, then their arcs are congruent. e. The diameter of a circle divides the circle into two congruent arcs.
2. apply the above theorems to numerical and algebraic problems, and to formal and informal proofs
Writing Exercise:
Amar created a pie chart of his personal expenses: rent, car payments, food, and entertainment. If the degree measure of the arc for rent was 60 degrees, what part of his pay check is spent on rent? Explain how you know this.

Student Name ____________________________
Unit 11-Aim 1: How can we find the measure of central angles?
1. “Johnny” is driving a car. The road is icy. “Johnny” turns the wheel, loses control and spins around one time. What is the name given to describe this rotation?
_____________________________________________________________________________________
2. What do we know about circles?
_____________________________________________________________________________________
3. An arc of a circle is part of the circle between two points on the circle. In the diagram below of circle
O , radii AO and CO are drawn such that : :

3: 4:5 . Find the measure of each arc.
4. In the diagram below of circle O , chords DF , DE , FG , and EG are drawn such that
: : :

5 : 2 :1: 7 . Find the measure of each arc.

5. A central angle of a circle is an angle whose vertex (point) is the center of the circle. a) Find the relationship between the central angles and their intercepted arcs (arcs between the angles). b) Find x . m BC
 m

BOC
90

90

 m

AOC
180

180
 m FC
 m FOC
60
 x
_____________________________________________________________________________________
_____________________________________________________________________________________
Fill in:
The degree measure of an arc measure of an arc is
_________to the measure of the central angle that intercepts the arc.

6. In the accompanying diagram of circle O , chord AB is parallel to chord CD . If , mAC

100 , find a) mBD b) c) Fill in:
The measures of the arcs intercepted between parallel chords of a circle are __________.
Remember: “hamburger”
Note: Numbers may be equal, but geometric figures may be congruent (same size and same shape).
7. In the accompanying diagram, CD is parallel to diameter AB . If m AC

25 , what is ?

8.
In the diagram of circle O below, chord CD is parallel to diameter
and mAC

30 .
What is mCD ?
1) 150 2) 120 3) 100 4) 60
9. In the diagram below of circle O , chord AB chord CD , and chord CD chord EF .
Which statement must be true?
1) CE

DF 2) AC

DF 3) AC

CE 4) EF

CD

Extra:
HW1 June 2011: 5 In the diagram below of circle O , is parallel to .
Which statement must be true?
(1)

(2)

(3)

(4)

HW1 Aug 2011: 29 In the diagram below, trapezoid ABCD, with bases and , is inscribed in circle O, with diameter . If , find .
HW1 Find the measure of the central angle that intercepts an arc with a degree measure of 35.
HW2 Find the measure of the central angle that intercepts an arc with a degree measure of 48.
HW3 Find the measure of the central angle that intercepts an arc with a degree measure of 90.
HW4 Find the measure of the central angle that intercepts an arc with a degree measure of 140.
HW4 Find the measure of the central angle that intercepts an arc with a degree measure of 180.
HW1 Find the measure of the arc intercepted by a central angle with a measure of 60 .

HW2 Find the measure of the arc intercepted by a central angle with a measure of 75 .
HW3 Find the measure of the arc intercepted by a central angle with a measure of 100 .
Find the measure of the arc intercepted by a central angle with a measure of 120 .
HW4 Find the measure of the arc intercepted by a central angle with a measure of 170 .
HW1 The endpoints of
are on circle O ,

89 , and

42 .
Find: , m AB , mBC , , mDA , mBCD , mDAB , ,
HW3 In the accompanying diagram of circle O , chord find

.
AB is parallel to chord CD . If m AC

120 ,
120
HW3 In the accompanying diagram of circle O , chord AB is parallel to chord CD . If

and mAB

80 , find

.

HW7 In the accompanying diagram of circle O , chord AB is parallel to chord CD . If

and m AB

70 , find

.
HW4 In the accompanying diagram, CD is parallel to diameter AB . If m AC

25 , what is ?
HW5 In the diagram below, trapezoid ABCD , with bases AB and
, is inscribed in circle O , with diameter
. If mAB

70 , find mBC .
70

HW8 In the diagram below of circle O , chord AB chord CD , and chord CD chord EF .
Which statement may be false?
1) CE

DF 2) AC

BD 3)

4) AC

DF
Test11 In the accompanying diagram of circle O , chord AB is parallel to chord CD . If

and mAB

65 , find

.
Test11 In the diagram below of circle O , chord AB chord CD , and chord CD chord EF .
Which statement may be false?
1) CE

BD 2) AC

BD 3)

4) CE

DF

June 2011: 5. In the diagram below of circle O , chord
is parallel to chord CD .
Which statement must be true?
1)
2)
3)
4)
Jan 2012: 29. In the diagram below, two parallel lines intersect circle O at points A , B , C , and D , with mAB x 20 and mDC

2 x

20 . Find m AB .