Before Section 3.3One more way to prove triangles congruent:

Angle-Angle-Side (AAS) Postulate

If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of a second triangle, then the two triangles are congruent.

X

Y

Z

A

B

C

If , and then .AB XY B Y C Z ABC XYZ

Given: ,

is the Midpoint of

Prove:

B C D F

M DF

BDM CFM

B

D F

C

M

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

Given

DM = MF Definition of midpoint

Given

Definition of congruent segments

∆BDM ∆CFM AAS

B C Given

D F

is the Midpoint of M DF

DM MF

3.3 CPCTC and Circles

Objectives: • Use CPCTC in proofs

• Know and use basic properties of circles

Congruent triangles: triangles whose corresponding parts are congruent.

This can be used in a proof only AFTER triangles have been proven congruent.

Corresponding Part of Congruent Triangles are Congruent.

Circle: the set of points in a plane equidistant from the one point, the center.

Notation: circle C or �C.

Radius: the distance from the center of the circle to any point on the circle.

C

Theorem 19: All radii of a circle are congruent.

Circumference of a Circle: C = 2r

Area of a Circle: A = r2

Example 1: Find the circumference and area of a circle with a radius of 12 units.

L

M

O

NExample 2:Given: O Prove: ∆NOL ∆NOM

Statements Reasons

1. O 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

OL OM ON All radii of a circle are congruent

NOL and NOM are right angles

NOL NOM

Definition of perpendicular

All right angles are congruent

∆NOL ∆NOM SAS

NO LM Given

Example 3:Given: Z is the midpoint of Y and W are complementary to VProve:

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

WZ ZY

Definition of midpointWZ = ZY

Y W

Definition of congruent

Given

Congruent Complements Theorem

∆VZY ∆XZW

Vertical Angles Theorem

ASA

V

Y

X

Z

WWY

VY WX

Z is the midpoint of WY

Y and W are comp. to V

VZY WZX

CPCTCVY WX

Example 4:Given: PProve:

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

All radii of a circle are congruent

Vertical Angles Theorem

SAS

CPCTC

KL MN

∆VZY ∆XZW

KL MN

LM

N

PK

KP LP NP MP

KPL NPM

Example 5:Given: Q RT = TSProve: TRQ TSQ

Statements Reasons

1. Q 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

All radii of a circle are congruent

SSS

CPCTC

∆TRQ ∆TSQ

QR QS QT

R S

Q

T

RT RS Definition of congruent

TRQ TSQ

GivenRT = RS

Example 6:Given: C is the midpoint of

AC = CEProve: ∆ABF ∆EDF

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

All right angles are congruent

Given

∆FCA ∆FCE

AC = CE

FC AEBD

AF FE

F

A ECB D

FC AE

FCA and FCE are right angles Definition of perpendicular

FCA FCE

FC FC Reflexive Property

AC CE Definition of congruent

SAS

CPCTC

Continued on next slide

A E

Example 6:Given: C is the midpoint of

AC = CEProve: ∆ABF ∆EDF

Statements Reasons

9. 9.

10. 10.

11. 11.

12. 12.

13. 13.

14. 14.

Definition of congruent

Given

∆ABF ∆EDF

FC AEBD

AF FE

F

A ECB D

C is the midpoint of BD

AB DE Subtraction Property

AF FE

SAS

BC = CD Definition of midpoint

BC CD

Given

Example 7:

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

SAS ∆AGD ∆BGE

and bisect each otherAE BD

Definition of bisect

AGD BGE Vertical Angles Theorem

AD BE CPCTC

Given: and bisect each other

and bisect each other

Prove:

AE BD

BF EC

AD FC

A

G H

CB

D FE

AG GE

BG GD Definition of bisect

Continued on next slide

Example 7:

Statements Reasons

7. 7. Given

8. 8.

9. 9.

10. 10.

11. 11.

12. 12.

13. 13.

SAS ∆BHE ∆CHF

and bisect each otherBF EC

Definition of bisect

BHE CHF Vertical Angles Theorem

FC BE CPCTC

Given: and bisect each other

and bisect each other

Prove:

AE BD

BF EC

AD FC

A

G H

CB

D FE

BH HF

EH HC Definition of bisect

AD FC Transitive Property