Proving Triangles Congruent
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Proving Triangles Congruent. G.6. Visit www.worldofteaching.com For 100’s of free powerpoints. F. B. A. C. E. D. The Idea of Congruence. Two geometric figures with exactly the same size and shape. How much do you need to know. . . . . . about two triangles

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Proving Triangles Congruent

G.6

Visit www.worldofteaching.com

For 100’s of free powerpoints.


F

B

A

C

E

D

The Idea of Congruence

Two geometric figures with exactly the same size and shape.


How much do you

need to know. . .

. . . about two triangles

to prove that they

are congruent?


B

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


SSS

SAS

ASA

AAS

HL

Do you need all six ?

NO !


B

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.


Included Angle

The angle between two sides

  • GHI

  • H

  • GIH

  • I

  • HGI

  • G

This combo is called

side-angle-side, or just SAS.


E

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.


Side-Angle-Side (SAS)

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.


Included Side

The side between two angles

GI

GH

HI

This combo is called

angle-side-angle, or just ASA.


E

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.


Angle-Angle-Side (AAS)

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!


Warning: No SSA Postulate

There is no such thing as an SSA postulate!

NOT CONGRUENT!


BUT: SSA DOES work in one

situation!

If we know that the two triangles are right triangles!

Side

Side

Side

Angle


We call this

HL,

for “Hypotenuse – Leg”

Remember! The triangles must be RIGHT!

Hypotenuse

Hypotenuse

Leg

RIGHT Triangles!

These triangles ARE CONGRUENT by HL!


Hypotenuse-Leg (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

  • ABHL

  • CB  GL

  • C andG 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!


Congruence Postulates

and Theorems

  • SSS

  • SAS

  • ASA

  • AAS

  • AAA?

  • SSA?

  • HL


Name That Postulate

(when possible)

SAS

ASA

SSA

AAS

Not enough info!


Name That Postulate

(when possible)

AAA

Not enough info!

SSS

SSA

SSA

Not enough info!

HL


Name That Postulate

(when possible)

Not enough info!

Not enough info!

SSA

SSA

HL

AAA

Not enough info!


Vertical Angles,

Reflexive Sides and Angles

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

Vertical Angles

Reflexive Side

side shared by two

triangles


Name That Postulate

(when possible)

Vertical Angles

Reflexive Property

SAS

SAS

Reflexive Property

Vertical Angles

SSA

AAS

Not enough info!


Reflexive Sides and Angles

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

Reflexive Side

side shared by two

triangles

Reflexive Angle

angle shared by two

triangles


Let’s Practice

ACFE

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

For ASA:

B D

For SAS:

AF

For AAS:


What s next
What’s Next

Try Some Proofs

End Slide Show


Choose a problem

End Slide Show

Choose a Problem.

Problem #1

SSS

Problem #2

SAS

Problem #3

ASA


Problem 4
Problem #4

AAS

Given

Vertical Angles Thm

Given

AAS Postulate


Problem 5
Problem #5

HL

Given ABC, ADC right s,

Prove:

Given

1. ABC, ADC right s

Given

Reflexive Property

HL Postulate


Congruence Proofs

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?


Given implies congruent parts

segments

angles

segments

angles

angles

Given implies Congruent Parts

midpoint

parallel

segment bisector

angle bisector

perpendicular



Step 1 mark the given

… and what it implies

Step 1: Mark the Given


Step 2 mark

Step 2: Mark . . .

… if they exist.


Step 3 choose a method
Step 3: Choose a Method

SSS

SAS

ASA

AAS

HL


Step 4 list the parts

STATEMENTS

REASONS

S

A

S

Step 4: List the Parts

… in the order of the Method


Step 5 fill in the reasons

STATEMENTS

REASONS

Step 5: Fill in the Reasons

S

A

S

(Why did you mark those parts?)


Step 6 is there more

STATEMENTS

REASONS

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

Step 6: Is there more?

S

A

S


Congruent Triangles Proofs

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
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
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 triangles1
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 triangles2
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 triangles3
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 triangles4
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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