Day 6: Triangle Congruence, Correspondence and Styles of Proof

Name: Day 6: Triangle Congruence, Correspondence and Styles of Proof Date: Geometry CC (M1D) Opening Exercise Given: CE bisects BD Statements 1. bisects 1.Given CE BD Reasons 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 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 3) SSS SSS 2) SAS SAS 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 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 3

STYLES OF PROOFS STYLE 1: STYLE 2: STYLE 3: STYLE 4: 4

Name: Day 7: Congruence Criteria for Triangles- SSS Date: 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 and onto. Determine and state whether C Z. Explain why. onto XY, BC onto YZ, ---------------------------------------------------------------------------------------------------------------------------------------------------------------- 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 Statements 1. AB BC 1. A D Reasons C 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. 5

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. 6

Example 3: In the diagram below, BD CA and 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: Day 6and7and8 LabLesson: Triangle Congruence, Correspondence and Styles of Proof Date: 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 L LM NM O M Method to Prove: MOL MON N 8

3) Given: CA DA B and E are right angles C D Method to Prove: ABC AED A B E 4) Given: B is the midpoint of EC E C A C Method to Prove: EBA CBD B E D 5) Given: AD CD, CB AB, AD CB A B Method to Prove: ΔABC ΔCDA D C 9

Problem Set: D A 6) Given: X is the midpoint of BD X BD bisects AC Method to Prove: DXC BXA C B 7) Given: In quadrilateral ABCD, diagonals AC bisect each other and BD A B Method to Prove: AEB CED E D C 8) Given: B D, AB CD A B Method to Prove: ABC CDA D C 10

9) Given: SUR TUR RU is the bisector of SRT R Method to Prove: SRU TRU S U T 10) Given: WXY is an isosceles triangle X XZ is the altitude Method to Prove: WXZ YXZ W Z Y Challenge: Can you write the Givens associated to the marked up diagram below? Givens: R S V 11 T U

Method to Prove: VSR VTU Finished? Try these 11) Given: C is the midpoint of BE AB BC DE EC AC DC A D Method to Prove: ABC DEC B C E 12) Given: bisects B D AC BAD B Method to Prove: ABC ADC A C D 13) Given: G is the midpoint of AD, AD bisects BE A B C Method to Prove: ABG DEG G F E D 12

14) Given: AB AD B CA bisects BAD C A Method to Prove: ABC ADC D 15) Given: E is the midpoint of AC A B ADE CBE E Method to Prove: AED CEB D C 16)Given: AB and CD intersect at E CD bisects AB AC CD BD DC B Method to Prove: ACE BDE C E D A 13

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Name: Day 8: Congruence Criteria for Triangles - SAS Date: 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) A Two pairs of sides and the included angle are congruent B B' C A' ---------------------------------------------------------------------------------------------------------------------------------------------------------------- 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? C 1) CBD 2) C E EDB 3) CDB EBD 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. B E D C' 18

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 19

Example 3: In the diagram below, JM KL, JM ML, KL ML, Prove that JML KLM Separate JML and 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 SEPARATE TRIANGLES (1) AC DB (3) ACD DBA (2) BA CD (4) BDA CAD 20

Name: Day 9: Congruence Criteria for Triangles- ASA Date: 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 ) 21

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 22

Example 2: In the diagram below, BF ^ AC, CE ^ AB, and AE @ AF Prove that ACE ABF and BF CE Separate triangles 23

Name: Day 9and10 LabLesson: LAB QUIZ REVIEW Date: Geometry CC (M1D) 1. James believes that he can prove these two triangles to be congruent using SSS. Is he correct? Explain your response. C D B E 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? Separate Triangles 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 24

6. Which triangle congruence criteria will determine congruence for given diagram? B E A) SSS B) SAS C) ASA D) AAS E) HL A C D 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? 25

9. In the diagram below of onto DE. AB ABC and DEF, a sequence of rigid motions maps AC onto DF, A onto D and Determine and state whether C F. Explain why. 10. In the diagram of ABC and DEC below, BAC EDC, and BE a) Prove that ACB DCE bisects AD b) Describe the rigid motion(s) that would map ACB to DCE. 26

11. In the diagram of ABD and CBD below, BD AC, and D is the midpoint of AC a) Prove that ADB CDB B A D C b) Precisely describe the rigid motion(s) that would map DABD onto DCBD 27

Name: Day 11: Triangle Congruence AAS Date: 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: Given: Given: Given: BD bisects, AD and BC bisect each, BY BA, and DAB DCB other C is the midpoint of BD XY CA ABC AC BD BC BX 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 28

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. 29

Example 3: In the diagram shown AD BC a) Prove that ADE BCE, AE BE and ADE BCE. 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 AP. Determine and state whether RW BW. Explain your answer. Separate Triangles 30

Name: Day 12: Triangle Congruence HL Date: 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 Explain your answer. A C. ------------------------------------------------------------------------------------------------------------------------------------------------- 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. 31

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? 32

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? 33

Name: Day 11and12 LabLesson: Triangle Congruence, Correspondence and Styles of Proof Date: Geometry CC (M1D) 34

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Name: Day 13: Triangle Congruence Addition/Subtraction Property Date: 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. Given that AC BD. We know that BC BC by the property. We know that BC BC by the property. So we can get AC BD by the postulate. So we can get by the postulate. 36

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 1. SY TZ 1. Given 2. YZ YZ 2. 3. SZ TY 3. Reasons Sutraction Property using sides (or angles) of triangles! Statements 1. SZ TY 1. Given 2. YZ YZ 2. 3. SY TZ 3. Reasons -------------------------------------------------------------------------------------------------------------------------------------------------- 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) 37

ACE and ABF Example 4: Given 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 38

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 1. DEFB 1. Given 2. AE DB, CF DB 2. Given 3. DFC BEA 3. Reasons 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. 39

Name: Day 14: Congruence Review for Test Date: 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 JKL DEF K E 2. In and, and KL EF. Which one additional statement could be used to prove that the two triangles are congruent using only the AAS AAS method? J D L F J F 1) 2) 3) 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 prove AMC 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. 40

ABC DEF 5. In the diagram below of and, a sequence of rigid motions maps AC onto DE. Determine and state whether ABC EDF. Explain why. AB onto DF, A onto D and 1) SAS SAS 2) SSS SSS 3) ASA ASA 4) These triangles are NOT congruent 6. As shown in the diagram below AB AE SAS SAS only? 1) 1 2 2) BC ED 3) ADB ACE 4) EC BD. Which piece of information is needed to prove ABD AEC by A B 1 2 E C D 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 AC 3) 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. 41

9. Complete the partial proof below for the accompanying diagram by providing reasons for any steps missing reasons. Given: DEFB Prove: EAB FCD, AE DB, CF DB, DE FB, DC AB 1. DEFB Statements 1. Given 2. AE DB, CF DB 2. Given 3. DFC BEA 3. Reasons 4. 5. 6. 7. DC AB DE BF EF EF DF BE 4. Given 5. Given 6. 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. 42

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? 43

Name: Day 13and 14 LabLesson: Triangle Congruence Date: Geometry CC (M1D) 44

Name: Day 15: Triangle Congruence Take Home Portion of Test Date: CC Geometry TAKE HOME PORTION OF TEST 6 POINT Due: FRIDAY 11/3 NO EXCEPTIONS In the diagram below,, and Prove that ABF CDE BF AC, DE AC. Bonus [2 possible points] Add two extra steps to the completed proof above that will prove that 45