congruent triangles

When we talk about congruent triangles,we mean everything about them is congruent. All 3 pairs of corresponding angles are equal….

And all 3 pairs of corresponding sides are equal

YZ

Corresponding partsWhen 2 figures are congruent the corresponding parts are congruent. (angles and sides)If Δ ABC is to Δ XYZ, which side is to BC?

For us to prove that 2 people are identical twins, we don’t need to show that all “2000” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 5 short cuts or methods.

SSSIf we can show all 3 pairs of corr.

sides are congruent, the triangles have to be congruent.

SASShow 2 pairs of sides and the

included angles are congruent and the triangles have to be congruent.

Includedangle

Non-includedangles

ASA, AAS

ASA – 2 anglesand the included side

A

SA

AAS – 2 angles andThe non-included side AA

S

HL ( hypotenuse leg ) is usedonly with right triangles, BUT,

not all right triangles.

HL ASA

Some other stuff we’ll use…..

This is called a common side.It is a side for both triangles.

We’ll use the reflexive property To show it is congruent to itself.

We’ll also use angle relationshipswe’ve learned to help us

Vertical angles Alt int angles

DEF OF MIDPOINT (will make sides congruent)DEF OF A BISECTOR (will make sides or angles congruent)

We’ll also use definition of bisector/midpoint

Steps to show congruence

1. Add any congruent marks forcommon sides, vertical anglesor alternate interior angles

2. See if there is enough info touse SSS, SAS, AAS, ASA or HL

3. There is no ASS or AAA!

Which method can be used toprove the triangles are congruent

Common side

SSS

Parallel linesalt int angles

Common side

SAS

Vertical angles

SAS

ExamplesIs it possible to prove the Δs are ?

)

)) (

((

No, there is no AAA thm!

))(

((

)

Yes, ASA

PROOFS

Follow all the steps we have done already but you will also need to add the GIVEN information onto the diagram

A

B

C

DE

1 2

Given: AB = BD EB = BC

Prove: ∆ABE ˜ ∆DBC=

SAS

A C

D

Given: AB = BD EB = BC

Prove: ∆ABE ˜ ∆DBC=B

E

1 2

SAS

AB = BD Given 1 = 2 Vertical anglesEB = BC Given∆ABE ˜ ∆DBC SAS=

STATEMENTS REASONS

A B

C

1 2

Given: CX bisects ACB A ˜ BProve: ∆ACX ˜ ∆BCX

X

==

AAS

CX bisects ACB Given 1 = 2 Def of angle bisc A = B Given CX = CX Reflexive Prop∆ACX ˜ ∆BCX AAS=