Proving Δs are : SSS, SAS, HL, ASA, &

AAS

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

E

D

F

A

B

C

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

EXAMPLE 1:

GUIDED PRACTICE

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

SOLUTION

1. ACB CAD

BC ADGIVEN :

PROVE : ACB CAD

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

YOUR TURN:

GUIDED PRACTICE

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

YOUR TURN (continued):

GUIDED PRACTICE

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 2.

YOUR TURN:

SASSide-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 BC YX, AC 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:

EXAMPLE 2

STATEMENTS REASONS

5. ABC CDA SAS Congruence Postulate

5.

Example 2 (continued):

Given: DR AG and AR GR

Prove: Δ DRA Δ DRG.

D

AR

G

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 (continued):

HLHypotenuse - 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 .

ASAAngle-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.

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 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.

A B

C

D

E

F

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 5 (continued):

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 6 (continued):

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:

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

Example 7 (continued):