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Name: ____________________________________________ Date:______________ Day 6: Triangle Congruence, Correspondence and Styles of Proof Geometry CC (M1D) Opening Exercise
Given: bisects
Statements Reasons
1. bisects 1.Given
2. 2.
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 _____________________________________________
CE BD
CE BD
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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 and below a sequence of rigid motions maps , AB onto DE ,
and Which method can be use to prove
b) Determine and state whether AC DF . Explain why.
ABC DEF A D,
B E. ABC DEF ?
1) AAS AAS 3) SSS SSS
2) SAS SAS 4) ASA ASA
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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 is congruent
to by only the SAS SAS criteria?
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
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STYLES OF PROOFS
STYLE 1:_________________________ STYLE 2:_________________________
STYLE 3:_________________________ STYLE 4:________________________
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Name: ____________________________________________________ Date:______________ Day 7: Congruence Criteria for Triangles- SSS Geometry CC (M1D) 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?
4. In the diagram below of and , a sequence of rigid motions maps AB onto XY , BC onto YZ ,
and onto .
Determine and state whether C Z . Explain why.
---------------------------------------------------------------------------------------------------------------------------------------------------------------- Side-Side-Side triangle congruence criteria (SSS): (All three sides are ≅) ---------------------------------------------------------------------------------------------------------------------------------------------------------------
Example 1: In the diagram below AB BC , D is the midpoint of AC
Prove that ABD CBD
Statements Reasons
1. AB BC 1.
2. D is the midpoint of AC 2.
3. AD DC 3.
4. BD BD 4.
5. ABD CBD 5.
b) Precisely describe the rigid motion(s) that would map ABD onto CBD .
B
A CD
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Example 2: In the diagram below, BD CD and 𝐸 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.
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Example 3: In the diagram below, BD CA and AB DC .
a) Prove that ABD DCA
b) Since we proved that the triangles are congruent what can we say about ABD and DCA ?
Separate Triangles
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L
M
N
O
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
__________
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A
D C
B
D
E
C
B
A
A
B
C
D
E
3) Given: CA DA
B and E are right angles
Method to Prove: ABC AED
__________
4) Given: B is the midpoint of EC E C
Method to Prove: EBA CBD
__________
5) Given: AD CD , CB AB ,
AD CB
Method to Prove: Δ ΔABC CDA
__________
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X
D
BC
A
D C
BA
Problem Set:
6) Given: X is the midpoint of BD
BD bisects AC Method to Prove: DXC BXA
__________
7) Given: In quadrilateral ABCD, diagonals AC and BD bisect each other Method to Prove: AEB CED
__________
8) Given: B D , AB CD
Method to Prove: ABC CDA
E
A
C
B
D
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R
S U T
T U
V
SR
X
W YZ
__________
9) Given: SUR TUR
RU is the bisector of SRT Method to Prove: SRU TRU
__________ 10) Given: WXY is an isosceles triangle
XZ is the altitude Method to Prove: WXZ YXZ
__________ Challenge: Can you write the Givens associated to the marked up diagram below? Givens:
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E
D
CB
A
D
C
B
A
FD
CA
E
B
G
Method to Prove: VSR VTU _____________
Finished? Try these
11) Given: C is the midpoint of BE
AB BC
DE EC
AC DC
Method to Prove: ABC DEC
__________
12) Given: AC bisects BAD
B D Method to Prove: ABC ADC
__________
13) Given: G is the midpoint of AD ,
AD bisects BE Method to Prove: ABG DEG
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C A
B
D
E
C
D
B
A
__________
14) Given:
ADAB
CA bisects BAD
Method to Prove: ABC ADC
__________
15) Given:
E is the midpoint of AC ADE CBE Method to Prove: AED CEB
__________
16)Given: AB and CD intersect at E
CD bisects AB
AC CD
BD DC
Method to Prove: ACE BDE
__________
E
A
C
B
D
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KAHOOT.IT
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IF TIME SECOND KAHOOT BASIC VOCABULARY
16
17
18
B'
C'
A'
A
B
C
Name: ____________________________________________________ Date:______________ Day 8: Congruence Criteria for Triangles - SAS Geometry CC (M1D) 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?
4. In AYB and ZYX below, AY ZY , AB XZ , and A Z . Which method proves AYB ZYX ?
(1) SSS SSS (3) AAS AAS
(2) SAS SAS (4) ASA ASA
---------------------------------------------------------------------------------------------------------------------------------------------------------------- Side-Angle-Side Triangle Congruence Criteria (SAS)
Two pairs of sides and the included angle are congruent ---------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. To be able to prove that BCD DEB by SAS, using the two given congruent corresponding sides, one piece of information is missing. Which of the following would be that piece of information?
1) CBD EDB 3) CDB EBD
2) C E 4) BD bisect CBE 2. Tiffany notices that two congruent corresponding sides and the corresponding angle and says that these two triangles are congruent by SAS. Is she correct? Explain.
E
BD
C
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Example 1:
a) Given: O is the midpoint of MP and NQ
Prove: MON POQ
b) Since the triangles are congruent what can we say about the remaining corresponding sides and angles?
c) Precisely describe the rigid motion(s) that would map MNO onto PQO
----------------------------------------------------------------------------------------------------------------------------------------------------------------
Example 2: In the diagram below, ,AB CD AB CD .
a) Prove that ABD CDB .
b) Since we proved the triangles are congruent what can we say about C and A ?
c) Precisely describe the rigid motion(s) that would map CDB onto ABD
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Example 3: In the diagram below, , ,JM KL JM ML KL ML ,
Prove that JML KLM
a) Since the triangles are congruent what can we say about MJL and LKM ?
b) Precisely describe the rigid motion(s) that would map JML onto KLM Example 4:
If BAD CDA and as shown in the diagram, what additional information would make ABD DCA using only
SAS SAS
(1) AC DB (3) ACD DBA
(2) BA CD (4) BDA CAD
Separate JML and KLM
SEPARATE TRIANGLES
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Name: _____________________________________________ Date:______________ Day 9: Congruence Criteria for Triangles- ASA Geometry CC (M1D) 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?
4. As shown in the diagram below, bisects . Which additional piece of information could be used to prove
by ASA ASA
(1) ABC ADC (3) ACB ACD
(2) BC DC (4) AB AD
5. Determine whether there is enough information to prove the two triangles below are congruent. If so, tell why the triangles are congruent and write a congruence statement.
6. In the diagram below, . Which statement can not be proven?
1) 2) 3) 4)
------------------------------------------------------------------------------------------------------------------------------------------------------------- Angle-Side-Angle triangle congruence criteria (ASA): (Two pairs of angles and the included side are ≅)
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Example 1: In the diagram below, M is the midpoint of HP, ÐH @ÐP .
Prove that HGM PRM and RP HG
b) Precisely describe the rigid motion(s) that would map DHGM onto DPRM
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Example 2: In the diagram below, BF ^ AC, CE ^ AB, and AE @ AF
Prove that ACE ABF and BF CE
Separate triangles
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Name: ____________________________________________ Date:______________ Day 9and10 LabLesson: LAB QUIZ REVIEW Geometry CC (M1D) 1. James believes that he can prove these two triangles to be congruent using SSS. Is he correct? Explain your response.
2. If and is the shortest side of , what is the shortest side of ?
A) B) C) D)
3. If ABC DEF . Which of the following statements is not necessarily true?
(1) ACB DEF (3) AB DE
(2) BCA EFD (4) AC DF
4. Given ACE and ABF shown in the diagram to the right, with AB AC . Which statement is needed to prove
DACE @ DABF by SAS SAS ?
1) ACE ABF 3) BF CE
2) AEC AFB 4) AF AE
5. In ABC and YBX below, BC BX , BY BA , and XY CA . Which method proves AYB ZYX ?
(1) SSS SSS (3) AAS AAS
(2) SAS SAS (4) ASA ASA
EB
DC
Separate Triangles
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6. Which triangle congruence criteria will determine congruence for given diagram? A) SSS B) SAS C) ASA D) AAS E) HL
7. In the diagram shown, BD bisects ÐABC which piece of additional information is needed to prove that
DABD@ DCBD by ASA ASA only.
1) BAD BCD 3) CDB ADB
2) BC BA 4) DC DA
8. Which picture shown does not contain enough information to prove that ABC DEF ?
AC
B
D
E
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9. In the diagram below of ABC and DEF , a sequence of rigid motions maps AC onto DF , A onto D and
AB onto DE .
Determine and state whether C F . Explain why.
10. In the diagram of ABC and DEC below, BAC EDC , and BE bisects AD
a) Prove that ACB DCE
b) Describe the rigid motion(s) that would map ACB to DCE .
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11. In the diagram of ABD and CBD below, BD AC , and D is the midpoint of AC
a) Prove that ADB CDB
b) Precisely describe the rigid motion(s) that would map DABD onto DCBD
B
A CD
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Name: ____________________________________________ Date:______________ Day 11: Triangle Congruence – AAS Geometry CC (M1D)
Opening Exercise: 1. Label the diagrams based on the given information then state the method you would use to prove the triangles shown are congruent.
Given:
BD bisects ABC , DAB DCB
Given:
AD and BC bisect each other
Given:
AC BD
C is the midpoint of BD
Given:
BC BX , BY BA , and
XY CA
Method:
Method:
Method:
Method:
2. If ABC DEF and BAC EDF as shown in the diagram, what additional information would make the
triangles congruent using only AAS AAS
(1) AC DF (3) AB DE
(2) BC EF (4) BAC EDF ---------------------------------------------------------------------------------------------------------------------------------------------------------------- Angle-Angle-Side (AAS): Two pairs of angles and a side that is not included are congruent
----------------------------------------------------------------------------------------------------------------------------------------------------------------
Example 1: In JKL and DEF , K E and KL EF . Which one additional statement could be used to prove
that the two triangles are congruent using only the AAS AAS method?
1) J D
2) L F
3) J F
4) JK DE
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Example 2: In the diagram shown BC EF , ÐB@ÐE , and AC DF .
a) Prove that ABC DEF and ÐA@ÐD
b) Describe a sequence of rigid motions that will map ABC onto DEF .
30
Example 3: In the diagram shown AD BC and ADE BCE .
a) Prove that ADE BCE , AE BE
b) Describe a single rigid motion that will map ADE onto BCE .
---------------------------------------------------------------------------------------------------------------------------------------------------------------
Example 4: In the diagram shown ∠𝐴 ≅ ∠𝑃, ∠𝐵 ≅ ∠𝑅, 𝑊 is the midpoint of 𝐴𝑃̅̅ ̅̅ . Determine and state whether
RW BW . Explain your answer.
Separate Triangles
31
Name: ____________________________________________ Date:______________ Day 12: Triangle Congruence –HL Geometry CC (M1D) 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?
4. In the diagram shown, E is the midpoint of , and . Determine and state whether A C . Explain your answer.
------------------------------------------------------------------------------------------------------------------------------------------------- Hypotenuse-Leg Triangle Congruence Criteria (HL)When two right triangles have congruent hypotenuses and a pair of congruent legs, then the triangles are congruent. -------------------------------------------------------------------------------------------------------------------------------------------------
Example 1: In the diagram below, it is known that AMB , CM @DM , CA^ AB and DB^ AB and M is the midpoint
of AB . Explain why AC must be congruent to BD.
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Example 2: In ABC and YBX below, YCB XBA , BY BA , and Y A .
a) Prove that AYB ZYX .
b) What sequence of rigid motions map ABC onto YBX ?
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Example 3: In the diagram below , ,BC CD AB AD CB BA .
a) Prove that DBCD @ DBAD
b) What single rigid motion maps ABD onto CBD ?
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Name: ____________________________________________ Date:______________ Day 11and12 LabLesson: Triangle Congruence, Correspondence and Styles of Proof Geometry CC (M1D)
35
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Name: ____________________________________________ Date:______________ Day 13: Triangle Congruence –Addition/Subtraction Property Geometry CC (M1D) 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?
4. In the diagram shown AC ^ DE, AD @CE , B is the midpoint of ED. Can we prove that ÐA@ÐC? Explain
your response.
5. Which statement is sufficient evidence that
1) There is a sequence of rigid motions that maps onto , onto EC . 2) AB DE , BC EC , and A D 3) ACB DCE , B E and A D
4) There is a sequence of rigid motions that maps onto , onto EC , and onto . ---------------------------------------------------------------------------------------------------------------------------------------------------------------
Addition Postulate Subtraction Postulate Given that . We know that BC BC by the ___________ property. So we can get AC BD by the ___________ postulate.
Given that AC BD . We know that BC BC by the ___________ property. So we can get by the ___________ postulate.
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Example 1: In the diagram of below, . Which reasons can be used to prove ?
1) reflexive property and subtraction postulate 2) transitive property and addition postulate 3) reflexive property and addition postulate 4) transitive property and subtraction postulate
Example 2: In the diagram below of , . Using this information, it could be proven that
(1) (2) (3) (4) ----------------------------------------------------------------------------------------------------------------------------------------------------------------
Applying Addition and Subtraction Postulate in Proofs
Addition Property using sides (or angles) of triangles!
Statements Reasons
1. SY TZ 1. Given
2. YZ YZ 2.
3. SZ TY 3.
Sutraction Property using sides (or angles) of triangles!
Statements Reasons
1. SZ TY 1. Given
2. YZ YZ 2.
3. SY TZ 3.
--------------------------------------------------------------------------------------------------------------------------------------------------
Example 3: In the diagram at the right , , ,BC AD ED AD AF CD A EFD .
Which method proves that ACB FDE ?
(1)ASA (Angle-Side-Angle) (2) AA (Angle-Angle) (3) SAS (Side-Angle-Side) (4) HL (Hypotenuse-Leg)
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Example 4: Given ACE and ABF shown in the diagram to the right, with AE AF . Which statement is needed to
prove ACE ABF by SAS SAS ?
1) ACE ABF
2) AEC AFB
3) BF CE
4) CF BE
Applying Addition and Subtraction Postulate in Proofs
Example 5: In the diagram shown , , ,AB BC DE EF BC EF AF DC
a) Prove that AB DE
Statements Reasons
1. ,AB BC DE EF 1. Given
2. ABC DEF 2.
3. BC EF 3. Given
4. AF DC 4. Given
5. FC FC 5.
6. AC DF 6.
7. ABC DEF 7.
8. AB DE 8.
b) Precisely describe a sequence of rigid motions that will map ABC onto DEF
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Practice NYTS(Now You Try Some!)
1. In the diagram below of , , Which reasons can be used to prove AB CD
1) reflexive property and subtraction postulate 2) transitive property and addition postulate 3) reflexive property and addition postulate 4) transitive property and subtraction postulate
2. In the diagram shown, DEFB , , , ,AE DB CF DB DE FB DC AB
Fill in the missing reasons to prove that EAB FCD
Statements Reasons
1. DEFB 1. Given
2. ,AE DB CF DB 2. Given
3. DFC BEA 3.
4. DC AB 4. Given
5. DE BF 5. Given
6. EF EF 6.
7. DF BE 7.
8. AEB CFD 8.
9. EAB FCD 9.
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Name: ________________________________________________________________ Date: __________
Day 14: Congruence Review for Test CC Geometry M1TD
1. In the diagram below of , AB CD . Which reasons can be used to prove that .
1) reflexive property and subtraction postulate 2) transitive property and addition postulate 3) reflexive property and addition postulate 4) transitive property and subtraction postulate
2. In JKL and DEF , K E and KL EF . Which one additional statement could be used to prove that the two triangles are congruent using only the AAS AAS method?
1) J D
2) L F
3) J F
4) JK DE
3. In the diagram shown it is given that CA^ AB and DB^ AB and M is the midpoint of AB . To proveAMC BMD by hypotenuse-leg only, which additional information would you need?
1) AC BD
2) CM @DM
3) AM BM 4) ACM BDM
4. Which statement is sufficient evidence that
1) There is a sequence of rigid motions that maps onto , onto EC . 2) AB DE , BC EC , and A D 3) ACB DCE , B E and A D
4) There is a sequence of rigid motions that maps onto , onto EC , and onto .
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5. In the diagram below of ABC and DEF , a sequence of rigid motions maps AC onto DF , A onto D and
AB onto DE . Determine and state whether ABC EDF . Explain why.
1) SAS SAS 2) SSS SSS 3) ASA ASA 4) These triangles are NOT congruent
6. As shown in the diagram below AB AE . Which piece of information is needed to prove ABD AEC by SAS SAS only? 1) 1 2
2) BC ED
3) ADB ACE
4) EC BD
7. AC bisects BAD and AC BD . Which of the following statements below is not true?
1) ABC ADC by ASA ASA
2) AC bisects BD
3) AC is an altitude of triangle ABD. 4) ABD is a scalene triangle.
8. In the diagram shown, E is the midpoint of , and . Determine and state whether . Explain your answer.
2
1
C
E
A D
B
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9. Complete the partial proof below for the accompanying diagram by providing reasons for any steps missing reasons.
Given: DEFB , , , ,AE DB CF DB DE FB DC AB
Prove: EAB FCD
Statements Reasons
1. DEFB 1. Given
2. ,AE DB CF DB 2. Given
3. DFC BEA 3.
4. DC AB 4. Given
5. DE BF 5. Given
6. EF EF 6.
7. DF BE 7.
8. AEB CFD 8.
9. EAB FCD 9.
10. In the diagram shown and bisect each other at E. a) Prove that DDEA @ DBEC . b) Describe a single rigid motion that maps DEA onto BEC .
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11. In ABC and YBX below, YCB XBA , BY BA , and Y A .
a) Prove that AYB ZYX .
b) What sequence of rigid motions map ABC onto YBX ?
44
Name: ____________________________________________ Date:______________ Day 13and 14 LabLesson: Triangle Congruence Geometry CC (M1D)
45
Name: __________________________________________________________ Date: __________ Day 15: Triangle Congruence Take Home Portion of Test CC Geometry
TAKE HOME PORTION OF TEST 6 POINT
Due: FRIDAY 11/3 NO EXCEPTIONS
In the diagram below , , and BF AC , DE AC .
Prove that ABF CDE
Bonus [2 possible points]
Add two extra steps to the completed proof above that will prove that
