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# 4.3  Δ s - PowerPoint PPT Presentation

4.3  Δ s. Objectives. Name and label corresponding parts of congruent triangles Identify congruence transformations.  Δ s. Triangles that are the same shape and size are congruent. Each triangle has three sides and three angles.

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### 4.3 Δs

• Name and label corresponding parts of congruent triangles

• Identify congruence transformations

Δs

• Triangles that are the same shape and size are congruent.

• Each triangle has three sides and three angles.

• If all six of the corresponding parts are congruent then the triangles are congruent.

• CPCTC –

Corresponding Parts of Congruent Triangles are Congruent

• Be sure to label Δs with proper mappings (i.e. if D  L, V  P, W  M, DV  LP, VW  PM, and WD  ML then we must write ΔDVW ΔLPM)

• Congruency amongst triangles does not change when you…

• slide,

• turn,

• or flip

• … one of the triangles.

So, we can only prove Δs  if ALL sides AND ALL s are .

NO!

• There are some shortcuts…

### 4.3 Proving Δs are  : SSS and SAS

• Use the SSS Postulate

• Use the SAS Postulate

Postulate 4.1 (SSS)Side-Side-Side  Postulate

• If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

SSS Postulate

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___

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If seg AB  seg ED, seg AC  seg EF, & seg BC  seg DF, then ΔABC ΔEDF.

B

C

___

___

E

___

___

___

___

___

___

D

F

Given: QR  UT, RS  TS, QS=10, US=10Prove: ΔQRS ΔUTS

U

Q

10

10

R

S

T

Statements Reasons

1. QR  UT, RS  TS,1. Given

QS=10, US=10

2. QS=US 2. Substitution

3. QS  US 3. Def of  segs.

4. Δ QRS Δ UTS 4. SSS Postulate

Postulate 4.2 (SAS)Side-Angle-Side  Postulate

• If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

• If seg BC  seg YX, seg AC  seg ZX, & C X, then ΔABC  ΔZXY.

B

Y

)

(

C

A

X

Z

Given: WX  XY, VX  ZX Prove: Δ VXW Δ ZXY

W

Z

X

1

2

Y

V

Statements Reasons

1. WX  XY; VX  ZX 1. Given

2. 1 2 2. Vert s Thm.

3. Δ VXW Δ ZXY 3. SAS Postulate

Given: RS  RQ and ST  QTProve: Δ QRT  Δ SRT.

S

Q

R

T

Statements Reasons

1. RS  RQ; ST  QT 1. Given

2. RT  RT 2. Reflexive

3. Δ QRT Δ SRT 3. SSS Postulate

Given: DR  AG and AR  GRProve: Δ DRA  Δ DRG.

D

R

A

G

1. DR  AG; AR  GR

2. DR  DR

3.DRG & DRA are rt. s

4.DRG   DRA

5. Δ DRG  Δ DRA

Reasons

1. Given

2. Reflexive Property

3.  lines form 4 rt. s

4. Right s Theorem

5. SAS Postuate

Proof

Pre-AP: Pg. 195 #9 – 16, 22 – 25 Pg. 204 #14 – 19, 22 – 25