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# Lesson 4.3 and 4.4 Proving Triangles are Congruent PowerPoint PPT Presentation

Lesson 4.3 and 4.4 Proving Triangles are Congruent. p. 212. Learning Target. I can list the conditions (SAS, SSS) to prove triangles are congruent. I can identify and use reflexive, symmetric and transitive property in my proof. How To Find if Triangles are Congruent.

Lesson 4.3 and 4.4 Proving Triangles are Congruent

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## Lesson 4.3 and 4.4 Proving Triangles are Congruent

p. 212

### Learning Target

I can list the conditions (SAS, SSS) to prove triangles are congruent.

I can identify and use reflexive, symmetric and transitive property in my proof.

### How To Find if Triangles are Congruent

• Two triangles are congruent if they have:

• exactly the same three sides and

• exactly the same three angles.

• But we don't have to know all three sides and all three angles ...usually three out of the sixis enough.

• There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

• ### 1. SSS   (side, side, side)

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

SSS stands for "side, side, side“

and means that we have two triangles

with all three sides equal.

For example:

is congruent to:

### 2. SAS (side, angle, side)

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

SAS stands for "side, angle, side"

and means that we have two triangles

where we know two sides and the included angle are equal.

For example:

is congruent to:

### 3. ASA   (angle, side, angle)

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

ASA stands for "angle, side, angle“

and means that we have two triangles

where we know two angles and the

included side are equal.

For example:

is congruent to:

### 4. AAS   (angle, angle, side)

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

AAS stands for "angle, angle, side“

and means that we have two triangles

where we know two angles and the

non-included side are equal.

For example:

is congruent to:

### 5. HL   (hypotenuse, leg)

HL applies only to right angled-triangles!

HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs")

and

### 5. HL   (hypotenuse, leg)

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

• It means we have two right-angled triangles with

• the same length of hypotenuseand

• thesame length for one of the other two legs.

• It doesn't matter which leg since the triangles could be rotated.

• For example:

is congruent to

### Caution ! Don't Use "AAA" !

Without knowing at least one side, we can't be sure if two triangles are congruent..

AAA means we are given all three

angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!

Because the triangles can have the same angles but be different sizes:

For example:

is congruent to

### Goal 2

Proving Triangles are Congruent

E

B

D

A

F

C

DEF

DEF

ABC

DEF

ABC

ABC

DEF

JKL

ABC

JKL

If  , then  .

L

If  and  , then .

J

K

You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent.

THEOREM

Theorem 4.4Properties of Congruent Triangles

Reflexive Property of Congruent Triangles

Every triangle is congruent to itself.

Symmetric Property of Congruent Triangles

Transitive Property of Congruent Triangles

Using the SAS Congruence Postulate

Prove that

AEBDEC.

1

2

1

2

Statements

Reasons

AE  DE, BE  CEGiven

1  2Vertical Angles Theorem

3

AEBDEC SAS Congruence Postulate

Proving Triangles Congruent

ARCHITECTURE You are designing the window shown in the drawing. You

want to make DRAcongruent to DRG. You design the window so that

DRAG and RARG.

D

A

G

R

GIVEN

DRAG

RARG

DRADRG

PROVE

MODELING A REAL-LIFE SITUATION

Can you conclude that DRADRG?

SOLUTION

Proving Triangles Congruent

GIVEN

RARG

DRADRG

PROVE

1

2

6

3

4

5

Statements

Reasons

Given

DRAG

If 2 lines are, then they form

4 right angles.

DRA and DRG

are right angles.

Right Angle Congruence Theorem

DRADRG

DRAG

Given

RARG

DRDR

Reflexive Property of Congruence

SAS Congruence Postulate

DRADRG

D

A

R

G

### Given: SP  QR; QP  PRProve  SPQ  SPR

S

StatementsReasons

1. Given

1. SP  QR; QP  PR

2. QPS and RPS are right ’s.

2. Def. of 

3. QPS  PRS

3. Rt.   Thm.

4. SP  SP

4. Reflexive POC

5.  SPQ  SPR

5. SAS  Post.

Q

R

P

### Pair-share

Work on classwork on “Congruence Triangle”

Sage and Scribe on #21 to #24