Module 5

Triangle Congruency

Criteria

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How  much  do  you                        need  to  know.  .  .                .  .  .  about  two  triangles                                to  prove  that  they                              are  congruent?    

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Do you need all six ?

NO !

5.4 SSS 5.3 SAS 5.2 ASA 6.2 AAS 6.3 HL

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1.  AB ≅ DE 2.  BC ≅ EF 3.  AC ≅ DF

ΔABC ≅ Δ DEF

B

A C

Side

Side

Side

The triangles are congruent by

SSS.

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

SIDE-­‐SIDE-­‐SIDE  (SSS)  

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The angle between two sides

Included Angle

∠  HGI ∠  G

∠  GIH ∠  I

∠  GHI ∠  H

This combo is called side-angle-side, or just SAS.

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Name the included angle:

YE and ES

ES and YS

YS and YE

Included Angle

S Y

E

∠ YES or ∠E ∠ YSE or ∠S ∠ EYS or ∠Y The other two

angles are the NON-INCLUDED

angles.

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1.  AB ≅ DE 2.  ∠A ≅ ∠ D 3.  AC ≅ DF

ΔABC ≅ Δ DEF

B

A C

included angle

Side

Angle Side

The triangles are congruent by

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.

SIDE-­‐ANGLE-­‐SIDE  (SAS)  

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The side between two angles

Included Side

GI HI GH

This combo is called angle-side-angle, or just ASA.

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Name the included side:

∠Y and ∠E

∠E and ∠S

∠S and ∠Y

Included Side

S Y

E

YE ES SY

The other two sides are the

NON-INCLUDED sides.

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1.  ∠A ≅ ∠ D 2.  AB ≅ DE 3.  ∠ B ≅ ∠ E

ΔABC ≅ Δ DEF

B

A C

included side

Angle

Side

Angle

The triangles are congruent by

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.

ANGLE-­‐SIDE-­‐ANGLE  (ASA)  

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1.  ∠A ≅ ∠ D 2.  ∠ B ≅ ∠ E 3.  BC ≅ EF

ΔABC ≅ Δ DEF

Non-included side

B

A C

Side Angle

Angle

The triangles are congruent

by 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.

ANGLE-­‐ANGLE-­‐SIDE  (AAS)  

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Warning: No SSA Postulate

There is no such thing as an SSA

postulate!

The triangles are NOTcongruent!

Side

Side

Angle

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Warning: No SSA Postulate

NOT CONGRUENT!

There is no such thing as an SSA

postulate!

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BUT: SSA DOES work in one situation!

If we know that the two triangles

are right triangles!

Side Side

Side

Angle 128 128

We call this

These triangles ARE CONGRUENT by HL!

HL, for “Hypotenuse – Leg”

Hypotenuse Leg

Hypotenuse

RIGHT Triangles!

Remember! The

triangles must be RIGHT!

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Hypotenuse-Leg (HL)

1. AB ≅ HL 2. CB ≅ GL 3. ∠C and ∠G

are rt. ∠ ‘s

ΔABC ≅ Δ DEF

The triangles are congruent

by HL.

Right Triangle

Leg

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.

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Warning: No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT!

Same Shapes!

Different Sizes!

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Congruence Postulates and Theorems

• SSS • SAS • ASA • AAS • AAA? • SSA? • HL

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Name That Postulate

SAS   ASA  

AAS  SSA  

(when possible)

Not enough info!

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Name That Postulate (when possible)

SSS  AAA  

SSA  

Not enough info!

Not enough info!

SSA  HL 134 134

Name That Postulate (when possible)

SSA  

AAA  

Not enough info!

Not enough info!

HL

SSA  

Not enough info!

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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

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Name That Postulate (when possible)

SAS  

AAS  

SAS  Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSA  

Not enough info!

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

Reflexive Side side shared by two

triangles Reflexive Angle angle shared by two

triangles

Reflexive Sides and Angles

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Let’s Practice Indicate 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

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Both triangles are congruent. ∠V ≅ ∠Y, WV ≅ XY, and WX ≅ WX by the Reflexive Property of Congruence. Although at first instance it looks like SSA, which is not a congruence postulate, you then realize that it is a RIGHT TRIANGLE, and it can be seen that we have a hypotenuse and a leg. Therefore, both triangles are congruent by the HL congruence postulate.  

ANSWER  

5.2/5.3/5.4/6.2/6.3  Classwork  PART  1

• GO  ONLINE  In  Class:  Due  in  TWO  Days?  Honors:    5.2:  3-­‐6,14,20  5.3  2-­‐7,12  5.4  4,6,14,15,16,17,23  6.2  1-­‐6-­‐9,21,27*  6.3  2,3,10-­‐11  Regular:  5.2:  3-­‐6,  5.3  2-­‐6,  5.4  4,6,14,15,  6.2  1-­‐6-­‐9,21  6.3  2,3,10    

Reminders:  q   Guess  what?!  A  quiz  is  nearing.      

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