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

!