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Chapter 3.3 CPCTC and Circles
Megan O’Donnell
9
5/30/08
Objectives
After studying this section you will be able to understand the following:
The principle of CPCTC The basic properties of circles
CPCTC
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent
C
P
C
T
C
CPCTC Explained
In the diagram Therefore, we must drawthe conclusion that This is because theangles are corresponding parts of congruent
triangles, meaning they are exact
replicasof each other.
DOGRIN R
IN D
OG
DR
The Basics of Circles
Point M is the center of the circle shown to the right. Circles are named by their center point. Thus, this circle is called Circle M.
M
Circle M
Radii of Circles
In a circle’s definition every point of the circle is equidistant from the center.
A line reaching from the center to a point on the outside of a circle, such as is called a radius.
E
L
LE
Theorem 19
Theorem 19 states that all radii of a circle are
A
LL
C
This means that LACA
Theorem 19
Sample Problem Using CPCTC
1. 1.Given
2. 2.Given
3. 3. If then
4. 4. Vertical angles are
5. 5. AAS (1,3,4)
6. 6. CPCTC
DA
32
CEBE
65 DCEABC
DEAE
Statement Reason
Given: ;
Prove: DEAE
DA 32
Sample Problem With Circles
RNQN
1. N 1.Given
2. 2.All radii of
a are
Statement Reason
NN
NP
R
M OL
Q
Given: N
Prove:
RNQN
As simple as
this!!
Sample Problem With Both Ideas
1. C 1. Given
2. 2.All radii of a
are
3. 3.Vertical angles
Are
4. 4.SAS (2,3,2)
5. 5. CPCTC
CACBCECD
Statement Reason
DECBCA
DECBCA
DEAB
DEAB
Given: C
Prove:
B
A
C
D
E
Extra Problems
1. 1.
2. 2.
3. 3.
4. 4.
WZWX VZVX WZVWXV
Given: ;
Prove:
W
X
V
ZY
Statement Reason
...More
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
MPRO
Statement Reason
M
R
OP
MPRO
PRMR Given: C ;
Prove:
And More!
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
DCAD
CBDABD
Statement Reason
A
D
C
B
Given: B
Prove:
DCAD
And Even More!!
LM
L
MN
Given: M
= 3x+5
=6x-4
Find: x
MN
!
!
!
!
!
! ! !
Answers
1. 1.Given
2. 2.Given
3. 3.Reflexive
4. 4.CPCTC
WZWX
VZVX
WVWV
WZVWXV
1. C 1.Given
2. 2.Given
3.Right s
3. Lines form right
4. 4.Rt s are
5. 5.All radii of a
are
6. 6.Reflexive
7. 7.SAS (4,5,6)
8. 8.CPCTC
MPRO
ROPROM
ROPROM
PRMR
RORO ROPROM
PRMR
And more Answers
1. C 1.Given
2. 2.Given
3. 3.All radii of a
Are
4. 4.Reflexive
5. 5.SSS (2,3,4)
6. 6.CPCTC
DCAD
CBAB BDBD
CDBADB
CBDABD
3x+5=6x-4
9=3x
X=3
We can set these segments equal to each other because they are radii. We learned that all radii of a circle are congruent.
Statement Reason
Works Cited
Fogiel, Matthew. Problem Solvers Geometry. Piscataway: Research and Education System, 2004.
Milauskas, George, Richard Rhoad, and Robert Whipple. Geometry for
Enjoyment and Challenge. Evanston: McDougal Littell,1991.
The end!
YAY GEOMETRY
!