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

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

More on the SSS Postulate

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

E

D

F

A

B

C

EXAMPLE 1 Use the SSS Congruence Postulate

Write a proof.

GIVEN KL NL, KM NM

Proof It is given that KL NL and KM NM

By the Reflexive Property, LM LN.

So, by the SSS Congruence Postulate, KLM NLM

PROVE KLM NLM

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

GUIDED PRACTICE for Example 1

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

SOLUTION

Yes. The statement is true.

1. DFG HJK

Side DG HK, Side DF JH,and Side FG JK.

So by the SSS Congruence postulate, DFG HJK.

GUIDED PRACTICE for Example 1

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

SOLUTION

2. ACB CAD

BC ADGIVEN :

PROVE : ACB CAD

PROOF: It is given that BC AD By Reflexive propertyAC AC, But AB is not congruent CD.

GUIDED PRACTICE for Example 1

Therefore the given statement is false and ABC is not Congruent to CAD because corresponding sides are not congruent

GUIDED PRACTICE for Example 1

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

SOLUTION

QT TR , PQ SR, PT TSGIVEN :

PROVE : QPT RST

PROOF: It is given that QT TR, PQ SR, PT TS. So bySSS congruence postulate, QPT RST. Yes the statement is true.

QPT RST 3.

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 If seg BC seg YX, seg AC

seg ZX, & C X, then ΔABC ΔZXY.B

A C X

Y

Z)(

EXAMPLE 2 Use the SAS Congruence Postulate

Write a proof.

GIVEN

PROVE

STATEMENTS REASONS

BC DA, BC AD

ABC CDA

1. Given1. BC DAS

Given2. 2. BC AD

3. BCA DAC 3. Alternate Interior Angles Theorem

A

4. 4. AC CA Reflexive Property of Congruence

S

EXAMPLE 2 Use the SAS Congruence Postulate

STATEMENTS REASONS

5. ABC CDA SAS Congruence Postulate

5.

Given: RS RQ and ST QT Prove: Δ QRT Δ SRT.

Q

R

S

T

Example 3:Example 3:

Statements Reasons________

1. RS RQ; ST QT 1. Given

2. RT RT 2. Reflexive

3. Δ QRT Δ SRT 3. SSS Postulate

Example 3:Example 3:RQ R

T

Given: DR AG and AR GR

Prove: Δ DRA Δ DRG.

D

AR

G

Example 4:Example 4:

Statements_______1. DR AG; AR GR2. DR DR3.DRG & DRA are rt.

s4.DRG DRA5. Δ DRG Δ DRA

Reasons____________1. Given 2. Reflexive Property3. lines form 4 rt. s

4. Right s Theorem

5. SAS PostulateD

A GR

Example 4:Example 4:

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

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

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

Given: A D, C F, BC EF

Prove: ∆ABC ∆DEF

Paragraph ProofParagraph Proof

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

A B

C

D

E

F

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: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:

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.

Proof:

Statements:1. BD BC2. AD ║ EC3. D C

4. ABD EBC

5. ∆ABD ∆EBC

Reasons:

1. Given

2. Given

3. If || lines, then alt. int. s are

4. Vertical Angles Theorem

5. ASA Congruence Postulate

Assignment

Geometry:Workbook pg 67 - 75