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A

C

E

F

D

Corresponding Parts

Previously we learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent.

- AB DE
- BC EF
- AC DF
- A D
- B E
- C F

ABC DEF

A

C

Side-Side-Side (SSS)

If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Side

E

Side

F

D

Side

- AB DE
- BC EF
- AC DF

ABC DEF

The triangles are congruent by SSS.

The angle between two sides

- GHI
- H

- GIH
- I

- HGI
- G

This combo is called

side-angle-side, or just SAS.

Y

S

Included Angle

Name the included angle:

YE and ES

ES and YS

YS and YE

YES or E

YSE or S

EYS or Y

The other two angles are the NON-INCLUDED angles.

If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.

included

angle

B

E

Side

F

A

Side

C

D

- AB DE
- A D
- AC DF

Angle

ABC DEF

The triangles are congruent by SAS.

The side between two angles

GI

GH

HI

This combo is called

angle-side-angle, or just ASA.

Y

S

Included Side

Name the included side:

Y and E

E and S

S and Y

YE

ES

SY

The other two sides are the NON-INCLUDED sides.

Angle-Side-Angle (ASA)

If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent.

included

side

B

E

Angle

Side

F

A

C

D

Angle

- A D
- AB DE
- B E

ABC DEF

The triangles are congruent by ASA.

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

B

Non-included

side

E

Angle

F

Side

A

C

D

Angle

- A D
- B E
- BC EF

ABC DEF

The triangles are congruent by AAS.

Warning: No SSA Postulate

There is no such thing as an SSA postulate!

Side

Angle

Side

The triangles are NOTcongruent!

BUT: SSA DOES work in one

situation!

If we know that the two triangles are right triangles!

Side

Side

Side

Angle

HL,

for “Hypotenuse – Leg”

Remember! The triangles must be RIGHT!

Hypotenuse

Hypotenuse

Leg

RIGHT Triangles!

These triangles ARE CONGRUENT by HL!

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

Right Triangle

Leg

Hypotenuse

- ABHL
- CB GL
- C andG are rt. ‘s

ABC DEF

The triangles are congruent by HL.

Warning: No AAA Postulate

There is no such thing as an AAA postulate!

Different Sizes!

E

Same Shapes!

B

A

C

F

D

NOT CONGRUENT!

(when possible)

Not enough info!

Not enough info!

SSA

SSA

HL

AAA

Not enough info!

Reflexive Sides and Angles

When two triangles touch, there may be additional congruent parts.

Vertical Angles

Reflexive Side

side shared by two

triangles

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

AAS

Not enough info!

When two triangles overlap, there may be additional congruent parts.

Reflexive Side

side shared by two

triangles

Reflexive Angle

angle shared by two

triangles

ACFE

Indicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

B D

For SAS:

AF

For AAS:

Problem #5

HL

Given ABC, ADC right s,

Prove:

Given

1. ABC, ADC right s

Given

Reflexive Property

HL Postulate

1. Mark the Given.

2. Mark …

Reflexive Sides or Angles/Vertical Angles

Also: mark info implied by given info.

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts …

in the order of the method.

5. Fill in the Reasons …

why you marked the parts.

6. Is there more?

angles

segments

angles

angles

Given implies Congruent Partsmidpoint

parallel

segment bisector

angle bisector

perpendicular

Step 1: Mark the Given

1. Mark the Given and what it implies.

2. Mark … Reflexive Sides/Vertical Angles

3. Choose a Method. (SSS , SAS, ASA)

4. List the Parts …

in the order of the method.

5. Fill in the Reasons …

why you marked the parts.

6. Is there more?

Using CPCTC in Proofs

- According to the definition of congruence, if two triangles are congruent, their corresponding parts (sides and angles) are also congruent.
- This means that two sides or angles that are not marked as congruent can be proven to be congruent if they are part of two congruent triangles.
- This reasoning, when used to prove congruence, is abbreviated CPCTC, which stands for Corresponding Parts of Congruent Triangles are Congruent.

Corresponding Parts of Congruent Triangles

- For example, can you prove that sides AD and BC are congruent in the figure at right?
- The sides will be congruent if triangle ADM is congruent to triangle BCM.
- Angles A and B are congruent because they are marked.
- Sides MA and MB are congruent because they are marked.
- Angles 1 and 2 are congruent because they are vertical angles.
- So triangle ADM is congruent to triangle BCM by ASA.

- This meanssides AD and BC are congruent by CPCTC.

Corresponding Parts of Congruent Triangles

- A two column proof that sides AD and BC are congruent in the figure at right is shown below:

Corresponding Parts of Congruent Triangles

- A two column proof that sides AD and BC are congruent in the figure at right is shown below:

Corresponding Parts of Congruent Triangles

- Sometimes it is necessary to add an auxiliary line in order to complete a proof
- For example, to prove ÐR @ ÐO in this picture

Corresponding Parts of Congruent Triangles

- Sometimes it is necessary to add an auxiliary line in order to complete a proof
- For example, to prove ÐR @ ÐO in this picture

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