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


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


ACBCAD

1.

GIVEN :

BC AD

ACBCAD

PROVE :

It is given that BC AD By Reflexive property

AC AC, But AB is not congruent CD.

PROOF:

YOUR TURN:

GUIDED PRACTICE

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

SOLUTION


YOUR TURN (continued):

GUIDED PRACTICE

Therefore the given statement is false and ABC is not

Congruent to CAD because corresponding sides

are not congruent


2.

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:

YOUR TURN:

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

B

Y

)

(

A

C

X

Z


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

Example 2:

Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE


EXAMPLE 2

Example 2 (continued):

STATEMENTS

REASONS

ABCCDA

SAS Congruence Postulate


Given dr ag and ar gr prove dra drg
Given: DR  AG and AR GRProve: Δ DRA  ΔDRG.

Example 4:

D

R

A

G


Example 4 (continued):

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


Theorem 4 5 hl hypotenuse leg theorem
Theorem 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:

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


Example 5 (continued):

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:

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


Example 6 (continued):

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:

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.


Example 7 (continued):

Reasons:

  • Given

  • Given

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

  • Vertical Angles Theorem

  • ASA Congruence Postulate

Statements:

  • BD  BC

  • AD ║ EC

  • D  C

  • ABD  EBC

  • ∆ABD  ∆EBC


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