2.2a: Exploring Congruent Triangles

M(G&M)–10–4 Applies the concepts of congruency by solving problems on or off a coordinate plane; or solves problems using congruency involving problems within mathematics or across disciplines or contexts.

GSE’s

G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent

G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

CCSS

Congruent triangles: Triangles that are the same size and the same shape.

A B

C

D E

F

In the figure DEF ABC

Congruence Statement: tells us the order in which the sides and angles are congruent

If 2 triangles are congruent:

The congruence statement tells us which parts of the 2 triangles are corresponding “match up”.

ABC DEF Means

A D, B E, C F

and

AB DE, BC EF,andAC DF

ORDER IS VERY IMPORTANT

3 Angles:

3 Sides:

A R

C

T E

F

In the figure TEF ARC

A T, R E, C F

AR TE, RC EF, AC TF

Example

Congruent Triangles

A

B C Y

Z

X

ZXYABC

RSTJKL

______J______S

______KL

RK

ST

Example 2

Write the Congruence Statement

Example 3

Example 3 : Congruence Statement

Finish the following congruence statement:

ΔJKL Δ_ _ _

K

J

L

M

N

N M L

Definition of Congruent: Triangles(CPCTC)

Two triangles are congruent if and only if their corresponding parts are congruent. (tells us when Triangles are congruent)

Are the 2 Triangles Congruent. If so writeThe congruence statement.

Ex. 2 Are these 2 triangles congruent? If so, write a congruence statement.

Reflexive Property

Does the Triangle on the left have anyof the same sides or angles as the triangle on the right?

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

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

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

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

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****

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