Mr. Wolf
Monday 12/15/08
Geometry
Grades 10-12
Unit 8: Circles
Arcs and Central Angles
Materials and Resources:
 Warm-up (1 per student)
 Arcs and Central Angles sheet (1 per student)
 protractors, rulers (1 pair per student)
 Exit Ticket (1 per student)
PA Standards Addressed:
Instructional Objectives:
 Students will be able to define and identify minor arcs, major acrs, central angles,
adjacent arcs, and congruent arcs.
 Students will be able to find the measure of central angles and arcs using
protractors.
 Students will be able to calculate the measure of central angles and arcs by
applying the Arc Addition Postulate.
Time
10 min
1 min
min
Activity
Warm-up
Agenda
Arcs and Central
Angles
Pg. 341 Classroom
Exercises #1-13
1 min
5 min
Agenda
Conclusion
Description
Pass out the Warm-up and review solutions.
Review the goals for the day.
Modeling:
Guiding:
Independent Practice:
Assessment:
Modifications:
Students with special needs…
Advanced students…
Revisit goals and identify whether they were met.
Pass out the Exit Ticket and collect at the bell.
Homework:
Pg. 341 Written Exercises #1-6, 10, 11
Lesson Reflection:
Geometry Fall 2008
Name: ________________________
Warm-up
Given CB and CD are tangent to circle A and m BCD  24 .
a. What is true about CB and CD ? Why is this true?
b. Find measures of DBC and BDC .
c. What is true about AB and AD ? Why is this true?
d. Find measures of ABC and ADC .
e. Find measures of ABD and ADB .
Geometry Fall 2008
Name: ________________________
Warm-up
Given CB and CD are tangent to circle A and m BCD  24 .
a. What is true about CB and CD ? Why is this true?
b. Find measures of DBC and BDC .
c. What is true about AB and AD ? Why is this true?
d. Find measures of ABC and ADC .
e. Find measures of ABD and ADB .
Geometry Fall 2008
Name: ________________________
Arcs and Central Angles
Definitions:
A central angle is an angle with its vertex at the center of a circle.
An arc is a segment on the perimeter of a circle with endpoints on a circle.
The measure of an arc is defined to be the measure of its central angle.
A minor arc is formed by two points on a circle with measure less than 180°.
A semicircle is formed by two points on a circle with measure equal to 180°.
A major arc is formed by two points on a circle with measure greater than 180°.
*Major arcs are named with three points whereas minor arcs are named with only two.
Activity 1: Circle A:
1) Draw radii AB and AC .
Conclusion Questions:
1) Using your protractor, measure the smaller part of BAC .
mBAC  ______
2) Minor arc BC has measure of ______.
3) Circles have a total of ______°.
4) Calculate the measure of the larger part of BAC .
5) Major arc BDC has measure of ______.
mBAC  ______
Definition:
Adjacent arcs of a circle are arcs that have exactly one point in common.
Activity 2: Circle R:
1) Draw radii RS , RT , and RU .
Conclusion Questions:
1) Using your protractor, measure the smaller part of SRT .
mSRT  ______
2) Minor arc ST has measure of ______.
3) Using your protractor, measure the smaller part of TRU .
mTRU  ______
4) Minor arc TU has measure of ______.
5) Without using your protractor, find the measure of SRU .
mSRU  ______
6) What postulate allows us to find the measure of SRU without using a
protractor?
7) Minor arc SU has measure of ______.
8) Major arc SWU has measure of ______.
9) What kind of arcs are ST and TU? What point do they share in common?
10) Complete the theorem:
The measure of the arc formed by two adjacent arcs is equal to _____________________
________________________________________________________________________
Definition:
Congruent arcs are arcs, in the same circle or in congruent circles, that have equal
measures.
Activity 3: Circle M:
1) Draw radii MP and MQ .
mPMQ  ______
2) Using your protractor, find the measure of PMQ.
3) Use your protractor to create another arc in the circle with the same measure as
PMQ. To do this, place your protractor on MP and mark a point on the circle at
the same measure as PMQ.
4) Label this point N.
Conclusion Questions:
1) Using your protractor, measure the smaller part of PMQ .
mPMQ  ______
2) Minor arc PQ has measure of ______.
3) Using your protractor, measure the smaller part of NMP .
mNMP  ______
4) Minor arc NP has measure of ______.
5) Without using your protractor, find the measure of NMQ .
mNMQ  ______
6) Minor arc NQ has measure of ______.
7) Major arc NQ has measure of ______.
8) Complete the theorem:
In the same circle or in congruent circles, two minor arcs are congruent if and only if
________________________________________________________________________
Definitions:
A central angle is an angle with its vertex at the center of a circle.
An arc is a segment on the perimeter of a circle with endpoints on a circle.
The measure of an arc is defined to be the measure of its central angle.
A minor arc is formed by two points on a circle with measure less than 180°.
A semicircle is formed by two points on a circle with measure equal to 180°.
A major arc is formed by two points on a circle with measure greater than 180°.
*Major arcs are named with three points whereas minor arcs are named with only two.
Adjacent arcs of a circle are arcs that have exactly one point in common.
Congruent arcs are arcs, in the same circle or in congruent circles, that have equal
measures.
Arc Addition Postulate – The measure of the arc formed by two adjacent arcs is the sum
of the measures of these two arcs.
Arc Congruency Theorem – In the same circle or in congruent circles, two minor arcs are
congruent if and only if their central angles are congruent.
Directions: Match the letter of the definition with the term it defines.
central angle
_____
arc
_____
measure of an arc
_____
minor arc
_____
semicircle
_____
major arc
_____
adjacent arcs
_____
congruent arcs
_____
Arc Addition Postulate
_____
Arc Congruency Theorem
_____
A. arcs that have exactly one point in common
B. formed by two points on a circle with measure
equal to 180°
C. In the same circle or in congruent circles, two
minor arcs are congruent if and only if their
central angles are congruent.
D. an angle with its vertex at the center of a circle
E. equal to the measure of its central angle
F. arcs in the same circle or in congruent circles that
have equal measures
G. The measure of the arc formed by two adjacent
arcs is the sum of the measures of these two arcs.
H. a segment on the perimeter of a circle with
endpoints on a circle
I. formed by two points on a circle with measure
greater than 180°
J. formed by two points on a circle with measure
less than 180°
Directions: Match the letter of the definition with the term it defines.
central angle
_____
arc
_____
measure of an arc
_____
minor arc
_____
semicircle
_____
major arc
_____
adjacent arcs
_____
congruent arcs
_____
Arc Addition Postulate
_____
Arc Congruency Theorem
_____
A. arcs that have exactly one point in common
B. formed by two points on a circle with measure
equal to 180°
C. In the same circle or in congruent circles, two
minor arcs are congruent if and only if their
central angles are congruent.
D. an angle with its vertex at the center of a circle
E. equal to the measure of its central angle
F. arcs in the same circle or in congruent circles that
have equal measures
G. The measure of the arc formed by two adjacent
arcs is the sum of the measures of these two arcs.
H. a segment on the perimeter of a circle with
endpoints on a circle
I. formed by two points on a circle with measure
greater than 180°
J. formed by two points on a circle with measure
less than 180°