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4.3 to 4.5 Proving Δ s are  : SSS, SAS, HL, ASA, & AAS. Objectives. Use the SSS Postulate Use the SAS Postulate Use the HL Theorem Use ASA Postulate Use AAS Theorem. Postulate 19 ( SSS ) Side-Side-Side  Postulate.

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4 3 to 4 5 proving s are sss sas hl asa aas

4.3 to 4.5 Proving Δs are  : SSS, SAS, HL, ASA, & AAS


Objectives
Objectives

  • Use the SSS Postulate

  • Use the SAS Postulate

  • Use the HL Theorem

  • Use ASA Postulate

  • Use AAS Theorem


Postulate 19 sss side side side postulate
Postulate 19 (SSS)Side-Side-Side  Postulate

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


More on the sss postulate

E

A

F

C

D

B

More on the SSS Postulate

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


4 3 to 4 5 proving s are sss sas hl asa aas

Write a proof.

GIVEN

KL NL,KM NM

PROVE

KLMNLM

Proof

KL NL andKM NM

It is given that

LM LN.

By the Reflexive Property,

So, by the SSS Congruence Postulate,

KLMNLM

EXAMPLE 1

Use the SSS Congruence Postulate


4 3 to 4 5 proving s are sss sas hl asa aas

DFGHJK

SideDG HK, SideDF JH,andSideFG JK.

So by the SSS Congruence postulate, DFG HJK.

for Example 1

GUIDED PRACTICE

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION

Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.

Yes. The statement is true.


4 3 to 4 5 proving s are sss sas hl asa aas

ACBCAD

2.

GIVEN :

BC AD

ACBCAD

PROVE :

It is given that BC AD By Reflexive property

AC AC, But AB is not congruent CD.

PROOF:

for Example 1

GUIDED PRACTICE

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION


4 3 to 4 5 proving s are sss sas hl asa aas

for Example 1

GUIDED PRACTICE

Therefore the given statement is false and ABC is not

Congruent to CAD because corresponding sides

are not congruent


4 3 to 4 5 proving s are sss sas hl asa aas

3.

QPTRST

GIVEN :

QT TR , PQ SR, PT TS

PROVE :

QPTRST

It is given that QT TR, PQ SR, PT TS.So by

SSS congruence postulate, QPT RST. Yes the statement is true.

PROOF:

for Example 1

GUIDED PRACTICE

Decide whether the congruence statement is true. Explain your reasoning.

SOLUTION


Postulate 20 sas side angle side postulate
Postulate 20 (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 .


More on the sas postulate
More on the SAS Postulate

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

B

Y

)

(

A

C

X

Z


4 3 to 4 5 proving s are sss sas hl asa aas

BC DA,BC AD

ABCCDA

STATEMENTS

REASONS

S

BC DA

Given

Given

BC AD

BCADAC

A

Alternate Interior Angles Theorem

S

ACCA

Reflexive Property of Congruence

EXAMPLE 2

Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE


4 3 to 4 5 proving s are sss sas hl asa aas

EXAMPLE 2

Use the SAS Congruence Postulate

STATEMENTS

REASONS

ABCCDA

SAS Congruence Postulate


Given rs rq and st qt prove qrt srt
Given: RS  RQ and ST  QT Prove: Δ QRT  Δ SRT.

Example 3:

S

Q

R

T


4 3 to 4 5 proving s are sss sas hl asa aas

R

Q

R

Example 3:

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 gr prove dra drg
Given: DR  AG and AR  GR Prove: Δ DRA  Δ DRG.

Example 4:

D

R

A

G


4 3 to 4 5 proving s are sss sas hl asa aas

Example 4:

Statements_______

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 Postulate

D

R

G

A


Theroem 4 5 hl hypotenuse leg theorem
Theroem 4.5 (HL)Hypotenuse - Leg  Theorem

  • If the hypotenuse and a leg of a right Δ are  to the hypotenuse and a leg of a second Δ, then the 2 Δs are .


Postulate 21 asa angle side angle congruence postulate
Postulate 21(ASA):Angle-Side-Angle Congruence Postulate

  • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.


Theorem 4 6 aas angle angle side congruence theorem
Theorem 4.6 (AAS): Angle-Angle-Side Congruence Theorem

  • If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.


Proof of the angle angle side aas congruence theorem
Proof of the Angle-Angle-Side (AAS) Congruence Theorem

Given: A  D, C  F, BC  EF

Prove: ∆ABC  ∆DEF

D

A

B

F

C

Paragraph Proof

You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B  E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.

E


Example 5
Example 5:

Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.


Example 51
Example 5:

In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. Thus, you can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.


Example 6
Example 6:

Is it possible to prove these triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.


Example 61
Example 6:

In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.


Example 7
Example 7:

Given: AD║EC, BD  BC

Prove: ∆ABD  ∆EBC

Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC. Use the fact that AD ║EC to identify a pair of congruent angles.


Proof
Proof:

Statements:

  • BD  BC

  • AD ║ EC

  • D  C

  • ABD  EBC

  • ∆ABD  ∆EBC

Reasons:

  • Given

  • Given

  • If || lines, then alt. int. s are 

  • Vertical Angles Theorem

  • ASA Congruence Postulate


Assignment
Assignment

  • Geometry:Workbook pg 67 - 75