4.4-4.5 & 5.2: Proving Triangles Congruent

p. 206-221, 245-251

http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt

Adapted from:

SSS - Postulate

If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)

Example #1 – SSS – Postulate

Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.

AC =

BC =

AB =

MO =

NO =

MN =

5

7 2 25 7 74

5

7 2 25 7 74

MONACB By SSS

Definition – Included Angle

K

J

L

K is the angle between JK and KL. It is called the included angle of sides JK and KL.

K

J

L

What is the included angle for sides KL and JL?

L

SAS - Postulate

QP

R

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS)

J

L

KS

AS

S

A

S

by SASPQRJKL

Example #2 – SAS – Postulate

S

N

L

W

K

Given: N is the midpoint of LW N is the midpoint of SK

Prove:

N is the midpoint of LWN is the midpoint of SK

Given

,LN NW SN NK Definition of Midpoint

LNS WNK Vertical Angles are congruent

SAS

WNKLNS

Statement Reason

1 1

2 2

3 3

4 4WNKLNS

Definition – Included Side

JK is the side between J and K. It is called the included side of angles J and K.

What is the included side for angles K and L?

KL

K

J

L

K

J

L

Z

XY

ASA - Postulate

K

J

L

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA)

by ASAZXYJKL

W

HA

KS

Example #3 – ASA – Postulate

AW WK

Given: HA || KS

Prove:

HA || KS, Given

HAW SKW Alt. Int. Angles are congruent

HWA SWK Vertical Angles are congruent

ASA Postulate

AW WKSKWHAW

SKWHAW

1 1

2 2

3 3

4 4

Statement Reasons

Note: is not Note: is not SSS, SAS, or ASA.SSS, SAS, or ASA.

Identify the Congruent Triangles.

Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.

ABC STRV V by SSSby SSS

PNO VUWV V

TSC

B

A

R

H I

J

K

M L P N

O

V W

U

by SASby SAS

JHIV

Example

Given:

Prove: MH HTStatement Reason

AHby bisected is MAT

MATex with vertisosceles is AMT

1) AHby bisected is MAT

MATex with vertisosceles is AMT

1) Given

AAS (Angle, Angle, Side)AAS (Angle, Angle, Side)

• If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . .

then the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

Example

Given:

Prove:

Statement Reason

WBofmidpoint theis

||

E

TBAW

TBAW

1) 1) WBofmidpoint theis

||

E

TBAW Given

2)2)

HL (Hypotenuse, Leg)HL (Hypotenuse, Leg)

• If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .

A

C

B

F

E

D

then the 2 triangles are

CONGRUENT!

***** only used with right triangles****

Example

Given:

Prove:Statement Reason

nglesright tria are CBD and ABD

right are BDC andBDA

CBAB

CA

1) 1) Given

2)2)

nglesright tria are CBD and ABD

right are BDC andBDA

CBAB

The Triangle Congruence The Triangle Congruence Postulates &TheoremsPostulates &Theorems

LAHALLHL

FOR RIGHT TRIANGLES ONLY

AASASASASSSS

FOR ALL TRIANGLES

Only this one is new

Summary

• Any Triangle may be proved congruent by: (SSS) (SAS)

(ASA)

(AAS)

• Right Triangles may also be proven congruent by HL ( Hypotenuse Leg)

• Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).

Example 1Example 1

F

E

D

A

C

B

? DF CB

if determine any way to thereis

diagram, in then informatio Given the

CPCTCby CB so

SASby CAB !YES!

DF

DEF

Example 2Example 2

• Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

A

C

B

F

E

D

No ! SSA doesn’t work

Example 3Example 3

• Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

D

A

C

B

YES ! Use the reflexive side CB, and you have SSS

Name That PostulateName That Postulate

SASSASASAASA

SSSSSSSSASSA

(when possible)

Name That PostulateName That Postulate(when possible)

ASAASA

SASASS

AAAAAA

SSASSA

Name That PostulateName That Postulate(when possible)

SASASS

SASSAS

SASASS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSSS

AA

Let’s PracticeLet’s PracticeIndicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

B D

For AAS: A F

AC FE

Homework Assignment