Warm Up

Given the line y = 2x + 10, and given the point (- 4, 5).– Write the equation of the line that is

perpendicular to the given line and goes through the given point.

– Write the equation of the line that is parallel.

Congruent TrianglesCongruent Triangles 4.1-4.2 4.1-4.2

Congruent TrianglesCongruent Triangles 4.1-4.2 4.1-4.2

Today’s Goals:Today’s Goals:1.1. To recognize congruent To recognize congruent

figures.figures.2.2. To prove two triangles To prove two triangles

congruent using SSS and congruent using SSS and SAS.SAS.

Proving Triangles Congruent

Proving Triangles Congruent

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Warm Up

Two geometric figures with exactly the same size and shape.

The Idea of a CongruenceThe Idea of a Congruence

A C

B

DE

F

Congruent Polygons

• If two polygons are congruent, then all the angles are congruent

• And all the sides are congruent.

Ex.1: Naming Congruent Parts

TJD RCF. List the congruent corresponding parts.

• Sides: TJ RC JD CF DT FR

• Angles: T R J C D F

T

J

DF

R

C

Third Angles Theorem

• If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

C F

Ex.2: Proving Triangles Congruent

Use the information given in the diagram. Give a reason why each statement is true.

a) PQ PS, QR SRb) PR PRc) Q S, QPR

SPRd) QRP SRPee PQR PSR

Given

Reflexive Property

Given

3rd Angles Thm.

Definition ofCongruent Triangles

How much do you How much do you need to know. . .need to know. . .

. . . about two triangles to prove that they are congruent?

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

Corresponding PartsCorresponding Parts

ABC DEF

B

A C

E

D

F

1. AB DE

2. BC EF

3. AC DF

4. A D

5. B E

6. C F

Do you need Do you need all six ?all six ?IsIs by definition the only way? by definition the only way?

NO !The Short Cuts

SSSSASASAAAS

Rigid• If you have the three sides then

there is no choice for the angles.• The triangle is rigid.• Remember how the 4,5,8 triangle

worked yesterday?

Side-Side-Side (SSS)Side-Side-Side (SSS)

1. AB DE

2. BC EF

3. AC DF

ABC DEF

B

A

C

E

D

F

Included Angle

• The angle between two sides (segments).

B is included between AB and CB. B

A C

The angle between two sides

Included AngleIncluded Angle

G I H

Name the included angle:

YE and ES

ES and YS

YS and YE

Included AngleIncluded Angle

SY

E

E

S

Y

Rigid?• When you hold a firm angle with

your hands is the distance between your fingertips fixed?

Pasta and Protractor• I taped pasta of lengths 5

inches and 3 inches on the protractor at a 25 degree angle. Hold the red pipe cleaner up for the third side. Do you have a choice as to how long the pipe cleaner can be?

Side-Angle-Side (SAS)Side-Angle-Side (SAS)

1. AB DE

2. A D

3. AC DF

ABC DEF

B

A

C

E

D

F

included angle

The side between two angles

Included SideIncluded Side

GI HI GH

Name the included side:

Y and E

E and S

S and Y

Included SideIncluded Side

SY

E

YE

ES

SY

I need four volunteers to demonstrate opposite and adjacent.

Rigid?• Do the lines have a fixed point of

intersection?• Try the pasta with angles.

Angle-Side-Angle-Side-AngleAngle (ASA) (ASA)

1. A D

2. AB DE

3. B E

ABC DEF

B

A

C

E

D

F

included

side

Angle-Angle-Side (AAS)Angle-Angle-Side (AAS)

1. A D

2. B E

3. BC EF

ABC DEF

B

A

C

E

D

F

Non-included

side

Just a short cut.• If you know two angles of a triangle

you can find the third.(They always add up to 180) Name that theorem.

• Thus, we are using ASA. This allows us to skip finding the other angle.

• ASA and AAS are the same.

Warning:Warning: No SSA Postulate No SSA Postulate

A C

B

D

E

F

NOT CONGRUENT

There is no such thing as an SSA

postulate!

Never ever say triangles are congruent by “donkey” forward or backwards!

Warning:Warning: No AAA Postulate No AAA Postulate

A C

B

D

E

F

There is no such thing as an AAA

postulate!

NOT CONGRUENT

You must have a side

to know the size!

The Congruence PostulatesThe Congruence Postulates

SSS correspondence

ASA correspondence

SAS correspondence

AAS correspondence

SSA correspondence

AAA correspondence

Name That PostulateName That Postulate

SASSASASAASA

SSSSSSSSASSA

(when possible)

Name That PostulateName That Postulate(when possible)

ASAASA

SASASS

AAAAAA

SSASSA

Sometimes the corresponding parts are not marked but you know they are congruent.

What should you look for?

Look for:

• Common Parts (Reflexive Property)• Vertical Angles• Angles formed by Parallel lines• Angles formed by Perpendicular Lines• Linear Pairs

• Substitution

Give the Supporting Fact Give the Supporting Fact thenthenName That PostulateName That Postulate(when possible)

SASASS

SASSAS

SASASS

Reflexive Property

Vertical Angles

Vertical Angles

Reflexive Property SSSS

AA

Warning!

•These extra facts are not reasons for the triangles to be congruent!

What are the reasons why two triangles can be

congruent?– Given– Definition– SSS– SAS– ASA– AAS and one more we will learn

tomorrow!

Try to Name That PostulateTry to Name That Postulate(when possible)

(when possible)Name That PostulateName That Postulate

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

Cool DownCool DownIndicate the additional information needed to enable us to apply the specified congruence postulate.

For ASA:

For SAS:

For AAS: