Exploring Congruent Triangles

Congruent triangles:

Triangles that are the same size and shape– Each triangle has six parts, three angles and three sides

– If the corresponding six parts of one triangle are congruent to

the six parts of another triangle, then the triangles are congruent • This is abbreviated by CPCTC (corresponding parts of congruent triangles are

congruent)

– Orientation of the triangles is not important. This means that

the triangles can be flipped, slid and turned around, and if the corresponding parts are congruent, the triangles are congruent

If ABC DEF, then the following parts are congruent:

Angles A D B E C F

Segments AB DE BC EF AC DF

E

D

B

AC

F

Segments: Angles:1. 1.2. 2.3. 3.

Note: The order matters. If ABC DEF, it is not the same as saying ABC FED.

EXAMPLE 1:

If ABC RQC, name the corresponding congruent sides and angles.

Congruent Sides– – –

Congruent Angles– – –

R

QB

A

C

Write the correct congruency statement.Compare the sides and the angles.

Example 2

3. 54.

• Do worksheet Parts of congruent triangles and exploring congruent triangles.

Congruence of triangles is:

Reflexive:• ABC ABC

C' A'

B'B

CA

Congruence of triangles is:

Symmetric• If ABC DEF, then DEF ABC

D

E

FC

A

B

Congruence of triangles is:

Transitive• If ABC DEF and DEF LMN, then ABC LMN.

G I

H

D

E

FC

A

B

Proving Triangle Congruency

There are 4 ways to prove that two triangles are congruent to each other. Remember, once you know that two triangles are congruent, then Corresponding Parts of Congruent Triangles are Congruent.

Side-Side-Side Postulate (SSS):

If all 3 sides of one triangle are congruent to all 3 sides of another triangle, then the two triangles are congruent.

Side-Angle-Side Postulate (SAS):

If two sides & the included angle (the angle between the two sides) of one triangle are congruent to two sides & the included angle of another triangle, then the two triangles are congruent.

Ex:

• Do worksheet ways to prove triangle congruence SAS SSS

D

E

FC

A

B

Angle-Side-Angle Postulate (ASA):

If 2 angles & the side between them in one triangle are congruent to 2 angles & the side between them in another triangle, then the 2 triangles are congruent.

Angle-Angle-Side Postulate (AAS):

If 2 angles & a side not between them in one triangle are congruent to 2 angles & the corresponding side not between them in another triangle, then the 2 triangles are congruent.

Determine which Postulate or theorem can be used to prove the 2 triangles are congruent. If it’s not possible, write Not Possible.

Remember, you can choose from SSS, SAS, ASA, or AAS.

Hypotenuse-Leg (HL)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangle are congruent.

Proofs:

G

F

E

I

H

Given : & G is the midpoint of Prove: if

HIFE EIandFHHIGFEG FHEI

Statements Reasons

1) Is the midpoint of

2)

3)

HIFE EIandFH

IGEGandHGFG

HIGFEG

Given:

Prove:

Q

R

S

T

TSRQ

TSRQ

STRQRT Statement Reasons

1)

2)

3)

4)

TSRQ

TSRQ

STRQRT

TRRT

STRQRT

Statement Reason

1. L is the Midpoint of

2.

3.

4.

5. ΔWRL ΔEDL

L

W

ER

D

Given: L is the Midpoint ofProve: ΔWRL ΔEDL

EDWRWE,

EDWRWE,

EW

ELWL

ELDWLR

Statement Reason

1.

2.

3.

4.

F

J

N

K

L M

Given:Prove:

JMKLNJMNKL ,

JMKLNJMNKL ,

MNLN

MNLN

NN LKNJMN

JK

LM

Statement Reason

1.

2. Is a right angle

3.

4. Is a right angle

5.

6.

7.

Given:Prove:

JKMLKLJMKMJK ,,

JKMLKLJMKMJK ,,

JKML

JKML

JKM

MLKM

LMKMKMK

LMKJMK