Name: Date: Day 6: Triangle Congruence, Correspondence and ...

[Pages:45]Name: ____________________________________________ Day 6: Triangle Congruence, Correspondence and Styles of Proof

Opening Exercise

Given: CE bisects BD

Statements

1. CE bisects BD

2.

1.Given 2.

Reasons

Date:______________ Geometry CC (M1D)

Define congruence in your own words:

Define congruence using your knowledge of basic rigid motions:

--------------------------------------------------------------------------------------------------------------------------------------------------In order to prove triangles are congruent, we do not need to prove all of their corresponding parts are congruent. Instead we will look at criteria that refer to fewer parts that will guarantee congruence. There are 5 ways to prove triangle congruence.

1. SAS SAS _____________________________________________

2. SSS SSS _____________________________________________

3. ASA ASA ____________________________________________ 4. AAS AAS ____________________________________________

5. HL HL _____________________________________________

Two sets of criteria that are NOT SUFFICIENT in proving triangles congruent are 1. AAA AAA_____________________________________________

2. SSA SSA _____________________________________________

1

Three things to look for when proving triangles congruent:

Vertical Angles

Reflexive Property (Shared Side)

Reflexive Property (Shared angle)

---------------------------------------------------------------------------------------------------------------------------------------------------------------Example 1: If they pairs of triangles below are congruent, then name their congruence criteria. (SSS, SAS, ASA, AAS, HL) If not, state that it is not sufficient to prove that the triangles are congruent.

Example 2: In the diagram of ABC and DEF below a sequence of rigid motions maps , AB onto DE , A D, and B E. Which method can be use to prove ABC DEF ?

1) AAS AAS 2) SAS SAS

3) SSS SSS 4) ASA ASA

b) Determine and state whether AC DF . Explain why. 2

Practice NYTS (Now you try some!)

1. Are the following pairs of triangles congruent? If they are, then name their congruence criteria.

(SSS, SAS, ASA, AAS, HL)

a) Yes / No __________

b) Yes / No __________

c) Yes / No __________

d) Yes / No __________

e) Yes / No __________

f) Yes / No __________

g) Yes / No __________

h) Yes / No __________

2. If you are given that

and

which additional statement is sufficient evidence that

to

by only the SAS SAS criteria?

is congruent

1) AC DF

2) 3) 4)

3. If you are given that

, and

which additional statement is sufficient evidence that

is

congruent to

by only the AAS AAS criteria?

1) AC DF

2)

3) AB DE 4) BC EF

4. In the diagram below of

and

, a sequence of rigid motions maps onto , onto , and

onto .

Are

and

congruent? If so, name the method.

Determine and state whether

. Explain why

3

STYLE 1:_________________________

STYLES OF PROOFS

STYLE 2:_________________________

STYLE 3:_________________________

STYLE 4:________________________

4

Name: ____________________________________________________ Day 7: Congruence Criteria for Triangles- SSS Opening Exercise

1. What are the 5 ways to prove triangles are congruent?

2. What do we use instead of SSA? When do we use it?

3. What 3 things do we check for when we are out of givens?

Date:______________ Geometry CC (M1D)

4. In the diagram below of and onto .

and

, a sequence of rigid motions maps AB onto XY , BC onto YZ ,

Determine and state whether C Z . Explain why.

---------------------------------------------------------------------------------------------------------------------------------------------------------------Side-Side-Side triangle congruence criteria (SSS): (All three sides are )

-------------------------------------------------------------------------------------------------------------------------B--------------------------------------

Example 1: In the diagram below AB BC , D is the midpoint of AC Prove that ABD CBD

1. AB BC

Statements

2. D is the midpoint of AC

3. AD DC

4. BD BD 5. ABD CBD

A

D

C

Reasons

1.

2.

3.

4.

5.

b) Precisely describe the rigid motion(s) that would map ABD onto CBD . 5

Example 2: In the diagram below, BD CD and is the midpoint of BC , prove that BED CED.

a) Since we proved that the triangles are congruent what can we say about EDB and EDC ?

b) Precisely describe the rigid motion(s) that would map one triangle onto the other. 6

Example 3: In the diagram below, BD CAand AB DC . a) Prove that ABD DCA

Separate Triangles

b) Since we proved that the triangles are congruent what can we say about ABD and DCA ? 7

Name: ____________________________________________

Date:______________

Day 6and7and8 LabLesson: Triangle Congruence, Correspondence and Styles of Proof Geometry CC (M1D)

Life's NOT FAIR Label Diagrams and choose method!

Warm Up: What are the 5 ways triangles in which we can prove triangles congruent:

_________ _________ _________ _________ _________

we can NOT use _____________ and ______________

Guided Practice: Use the given statements to help you mark up the diagrams accordingly. State the method to prove the triangles are congruent. [Diagrams are not drawn to scale]

1) Given: IE GH , EF HF , F is the midpoint of GI

Method to Prove: EFI HFG

__________

2) Given: OM bisects LMN LM NM

Method to Prove: MOL MON __________

L O

N

M

8

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