4.3 4.3 ΔΔss

ObjectivesObjectives

Name and label corresponding parts Name and label corresponding parts of congruent trianglesof congruent triangles

Identify congruence transformationsIdentify congruence transformations

ΔΔss

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

Each triangle has three sides and Each triangle has three sides and three angles.three angles.

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

CPCTCCPCTC

CPCTC – CPCTC –

Corresponding Parts of Corresponding Parts of Congruent Triangles are Congruent Triangles are CongruentCongruent

Be sure to label Be sure to label ΔΔs with s with proper mappings (i.e. if proper mappings (i.e. if D D L, L, V V P, P, W W M, M, DV DV LP, VW LP, VW PM, and PM, and WD WD ML then we must write ML then we must write ΔΔDVWDVW ΔΔLPM)LPM)

Congruence Congruence TransformationsTransformations

Congruency amongst triangles does Congruency amongst triangles does not change when you…not change when you…

slide, slide, turn, turn, or flip or flip … … one of the triangles.one of the triangles.

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

NO!NO!

There are some shortcuts…There are some shortcuts…

4.3 Proving 4.3 Proving ΔΔs are s are : SSS : SSS and SASand SAS

ObjectivesObjectives

Use the SSS Postulate Use the SSS Postulate Use the SAS PostulateUse the SAS Postulate

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

If 3 sides of one If 3 sides of one ΔΔ are are to to 3 sides of another 3 sides of another ΔΔ, then , then the the ΔΔs are s are ..

SSS PostulateSSS Postulate

If seg AB If seg AB seg ED, seg ED, seg AC seg AC seg EF, & seg EF, & seg BC seg BC seg DF, seg DF, then then ΔΔABC ABC ΔΔEDF.EDF.

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A

B CE

D F

GivenGiven: QR : QR UT, RS UT, RS TS, QS=10, TS, QS=10, US=10US=10

ProveProve: : ΔΔQRS QRS ΔΔUTSUTS

Q

R S T

U

10 10

ProofProof

Statements ReasonsStatements Reasons

1. 1. QR QR UT, RS UT, RS TS, TS, 1. Given1. Given

QS=10, US=10QS=10, US=10

2. QS=US 2. Substitution2. QS=US 2. Substitution

3. QS 3. QS US US 3. Def of 3. Def of segs. segs.

4. 4. ΔΔ QRS QRS ΔΔ UTS 4. SSS Postulate UTS 4. SSS Postulate

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

If 2 sides and the If 2 sides and the included included of one of one ΔΔ are are to 2 sides and to 2 sides and the the included included of another of another ΔΔ, , then the 2 then the 2 ΔΔs are s are ..

If seg BC If seg BC seg YX, seg AC seg YX, seg AC seg ZX, & seg ZX, & C C X, then X, then ΔΔABC ABC ΔΔZXY.ZXY.B

AC

X

Y

Z)(

SAS PostulateSAS Postulate

Given: WX Given: WX XY, VX XY, VX ZX ZX Prove: Prove: ΔΔ VXW VXW ΔΔ ZXY ZXY

1 2

W

V

X

Z

Y

ProofProof

Statements ReasonsStatements Reasons

1. WX 1. WX XY; VX XY; VX ZX ZX 1. Given 1. Given

2. 2. 1 1 2 2. Vert 2 2. Vert s Thm.s Thm.

3. 3. ΔΔ VXW VXW ΔΔ ZXY 3. SAS ZXY 3. SAS PostulatePostulate

Given: RS Given: RS RQ and ST RQ and ST QT QTProve: Prove: ΔΔ QRT QRT ΔΔ SRT. SRT.

Q

R

S

T

ProofProof

Statements ReasonsStatements Reasons

1. RS 1. RS RQ; ST RQ; ST QT QT 1. Given 1. Given

2. RT 2. RT RT RT 2. Reflexive 2. Reflexive

3. 3. ΔΔ QRT QRT ΔΔ SRT SRT 3. SSS 3. SSS PostulatePostulate

Given: DR Given: DR AG and AR AG and AR GR GRProve: Prove: ΔΔ DRA DRA ΔΔ DRG. DRG.

D

AR

G

ProofProofStatementsStatements

1. DR 1. DR AG; AR AG; AR GR GR

2. DR 2. DR DR DR

3.3.DRG & DRG & DRA are DRA are rt. rt. ss

4.4.DRG DRG DRA DRA

5. 5. ΔΔ DRG DRG ΔΔ DRA DRA

ReasonsReasons1. Given 1. Given 2. Reflexive Property2. Reflexive Property3. 3. lines form 4 rt. lines form 4 rt. s s

4. Right 4. Right s Theorem s Theorem

5. SAS Postuate5. SAS Postuate

AssignmentAssignment

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