Warm-upWarm-upUse S for sides and A for angles to Use S for sides and A for angles to

describe the order of the congruent describe the order of the congruent parts in the triangles below.parts in the triangles below.

1.

2.

3.

4.

5.

Proving Proving ΔΔs are s are : : SSS and SASSSS and SAS

Outcome #11 – Logic & Geometric Outcome #11 – Logic & Geometric ProofsProofs

I can prove triangles congruent I can prove triangles congruent using SSS, SAS, ASA Congruence using SSS, SAS, ASA Congruence Postulates and AAS Congruence Postulates and AAS Congruence

Theorem.Theorem.

ObjectivesObjectives

I can prove two triangles are I can prove two triangles are congruent using SSS congruent using SSS Postulate and Postulate and SAS SAS Postulate Postulate

I can complete a two-column proof I can complete a two-column proof using SSS using SSS Postulate and SAS Postulate and SAS PostulatePostulate

Remember?Remember?

As of yesterday, As of yesterday, ΔΔs could only s could only be be if ALL sides AND angles if ALL sides AND angles were were

NOT ANY MORE!!!!NOT ANY MORE!!!! There are two short cuts to add.There are two short cuts to add.

Side-Side-Side (SSS) Side-Side-Side (SSS) Congruence PostulateCongruence Postulate

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

Meaning:Meaning:

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

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A

B CE

D F

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

ProveProve: : ΔΔQRS QRS ΔΔUTSUTS

Q

R S T

U

10 10

ProofProofStatements ReasonsStatements Reasons

1. 1. Given1. 1. Given

2. QS=US 2. Subs. Prop. 2. QS=US 2. Subs. Prop. of =of =

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

4. 4. ΔΔ QRS QRS ΔΔ UTS 4. SSS UTS 4. SSS PostulatePostulate

Side-Angle-Side (SAS)Side-Angle-Side (SAS)Congruence Postulate Congruence Postulate

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

If seg BC If seg BC seg YX, seg AC seg YX, seg AC seg ZX, and seg ZX, and C C X, X, then then ΔΔABC ABC ΔΔZYX.ZYX.

B

AC

X

Y

Z)(

Given: seg WX Given: seg WX seg. XY, seg. XY, seg VX seg VX seg ZX, seg ZX,

Prove: Prove: ΔΔ VXW VXW ΔΔ ZXY ZXY

1 2

W

V

X

Z

Y

ProofProof

Statements ReasonsStatements Reasons

1. seg WX 1. seg WX seg. XY seg. XY 1. Given 1. Given seg. VX seg. VX seg ZX seg ZX

2. 2. 1 1 2 2. Vert 2 2. Vert s s ThmThm

3. 3. ΔΔ VXW VXW ΔΔ ZXY 3. SAS ZXY 3. SAS Post.Post.

Given: seg RS Given: seg RS seg RQ and seg RQ and seg ST seg ST seg QT seg QT

Prove: Prove: ΔΔ QRT QRT ΔΔ SRT. SRT.Q

R

S

T

ProofProof

Statements ReasonsStatements Reasons

1. Seg RS 1. Seg RS seg RQ seg RQ 1. Given 1. Given seg ST seg ST seg QT seg QT

2. Seg RT 2. Seg RT seg RT seg RT 2. Reflex 2. Reflex prop prop

3. 3. ΔΔ QRT QRT ΔΔ SRT SRT 3. SSS 3. SSS postpost

Given: seg DR Given: seg DR seg AG and seg AG and seg AR seg AR seg GR seg GR

Prove: Prove: ΔΔ DRA DRA ΔΔ DRG. DRG.

D

AR

G

ProofProofStatementsStatements

1.1. seg DR seg DR seg AG seg AG

seg AR seg AR seg GR seg GR

2. seg DR 2. seg DR seg DR seg DR

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

4.4.DRG DRG DRA DRA

5. 5. ΔΔ DRG DRG ΔΔ DRA DRA

ReasonsReasons1.1. Given Given

2.2. Reflex. Prop of Reflex. Prop of 3.3. lines form 4 rt. lines form 4 rt. s s

4. Rt. 4. Rt. s thm s thm

5. SAS 5. SAS post. post.