Testing for Congruent Triangles Examples

Transcription

1 Testing for Congruent Triangles Examples 1. Why is congruency important? In 1913, Henry Ford began producing automobiles using an assembly line. When products are mass-produced, each piece must be interchangeable, so they must have the same size and shape. Each piece is an exact copy of the others, and any piece can be made to coincide with the others.. Student activity Have students draw a triangle and cut it out. Use it as a pattern to draw a second triangle and cut that triangle out. If one triangle is placed on top of the other, the two coincide or match exactly. This means that each part of the first triangle matches exactly the corresponding part of the second triangle. You have made a pair of congruent triangles. 3. If ABC is congruent to RST ( ABC RST), the vertex labeled A corresponds to the vertex labeled R, vertex B corresponds to S, and vertex C corresponds to T. This correspondence can be described in terms of angles and sides as follows. A corresponds to R B corresponds to S C corresponds to T AB corresponds to RS BC corresponds to ST AC corresponds to RT Since the two triangles match exactly, the corresponding parts are congruent. 4. Definition of Congruent Triangles (CPCTC) Two triangles are congruent if and only if their corresponding parts are congruent. The abbreviation CPCTC means Corresponding Parts of Congruent Triangles are Congruent. Testing for Congruent Triangles October 8, 001 1

2 5. Example A triangular wedge is used to anchor the seat belts of a car. a. Draw two identical wedges and label the vertices P, R, and S on one part and K, L, and M on the other so that PRS KLM. Then mark the congruent parts. b. What angle in PRS is congruent to K in KLM? P is congruent to K c. Which side of KLM is congruent to PS in PRS? KM is congruent to RS Point out the importance of the order of the letters in a statement of congruence. PRS KLM, but PRS is not congruent MLK. 6. Example The Adams family is having their game room renovated. This room will have two triangular windows. a. Draw two identical windows and label the vertices ABC on one window and DEF on the other so that ABC DEF. b. What angle in ABC is congruent to F in DEF? C c. Which side of DEF is congruent to AC in ABC? DF 7. Congruence of triangles, like congruence of segments and angles, is reflexive, symmetric, and transitive. Theorem Congruence of triangles is reflexive, symmetric, and transitive. Testing for Congruent Triangles October 8, 001

3 a. Prove that congruence of triangles is reflexive. : XYZ Prove : XYZ XYZ Statements XYZ X X, Y Y, Z Z XY XY, YZ YZ, XZ XZ XYZ XYZ Reasons Congruence of s is reflexive. Congruence of segments is reflexive. Definition of congruent triangles b. Prove that congruence of triangles is Symmetric. Statements Reasons LMN OPQ L O, M P, N Q, LM OP, CPCTC MN PQ, LN OQ O L, P M, Q N Congruence of angles is symmetric. OP LM, PQ MN, OQ LN Congruence of segments is symmetric. OPQ LMN Definition of congruent triangles. c. Prove that congruence of triangles is transitive. : ABC DEF, DEF GHI Prove ABC GHI Paragraph Proof: We are given that ABC DEF. By the definition of congruent triangles, the corresponding parts of the triangles are congruent. So, A D, B E, C F, AB DE, BC EF, and AC DF. It is also given that DEF GHI, so by the definition of congruent triangles, D G, E H, F I, DE GH, EF HI, and DF GI. Since congruence of angles is transitive, A G, B H, C I. Congruence of segments is transitive, so AB GH, BC HI, and AC GI. Therefore, ABC GHI by the definition of congruent triangles. Testing for Congruent Triangles October 8, 001 3

4 8. Side-Side-Side (SSS) Postulate If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. The SSS postulate can be used to prove triangles congruent. 9. : AB DB, AC CD Prove: ABC DBC Statements Reasons AB DB AC CD BC CB Common Side (Reflexive Property) ABC DBC SSS 10. Example ABC with vertices A(0, 5), B(, 0), and C(0, 0) and RST with vertices R(5, 8), S(5, 3), T(3, 3), show that ACB RST. Use the distance formula to show that the corresponding sides are congruent. AC = ( 0 0) + (5 0) AC = 5 = 5 RS = ( 5 5) + (8 3) RS = 5 = 5 AB = ( 0 ) + (5 0) AB = 9 RT = ( 5 3) + (8 3) RT = 9 CB = ( 0 ) + (0 0) CB = 4 = ST = ( 5 3) + (3 3) ST = 4 = All the pairs of corresponding sides are congruent, so ACB RST by SSS. Testing for Congruent Triangles October 8, 001 4

5 11. Example PQR with vertices P(3, 4), Q(, ), and R(7, ) and STU with vertices S(6, -3), T(4, -), and U(4, -7), show that PQR STU. Use the distance formula to show that the corresponding sides are congruent. PQ = PQ = 5 PR = ( 3 ) + (4 ( 3 7) + (4 PR = 0 = 5 QR = QR = 5 = 5 TU = TU = 5 = 5 ST = ST = 5 SU = ( 7) + ( ) ) ) ( 4 4) + ( ( ( 6 4) = ( 3 ( ( 6 4) + ( 3 ( 7)) )) 7)) SU = 0 = 5 All the pairs of corresponding sides are congruent, so PQR STU by SSS. 1. Side-Angle-Side (SAS) Postulate If two sides and the included angle of one triangle are congruent to two sides and an included angle of another triangle, then the triangles are congruent. 13. Example: : WZ YZ, VZ ZX Prove: VZW XZY Statements Reasons WZ YZ, VZ ZX WZV YZX Vertical s are VZW XZY SAS Testing for Congruent Triangles October 8, 001 5

6 14. Example: : AC CD, BC CE Prove: ABC DEC Statements Reasons AC CD BC CE (Included Angle) ACB DCE Vertical Angles ABC DEC SAS 15. Example: : 1 and are right angles, ST TP Prove: 3 4 Statements Reasons 1 and are right angles, ST TP 1 Al rt s are TR TR Congruence of segments is reflexive STR PTR SAS 3 4 CPCTC 16. Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Testing for Congruent Triangles October 8, 001 6

7 17. Example: : Q and S are right angles, QR SR Prove: P T Statements Reasons QR SR PRQ TRS Q and S are right angles Q S PQR TRS P T Vertical s are All rt. s are ASA CPCTC 18. Example: : BE bisects AD, A D Prove: AB DC Statements Reasons BE bisects AD, A D 1 Vertical s are AE ED Def. of bisector AEB DEC ASA AB DC CPCTC Testing for Congruent Triangles October 8, 001 7

8 Name: Date: Class: Testing for Congruent Triangles Worksheet 1. Draw triangles TLA and RSB. Mark the corresponding parts for TLA RSB.. Describe how you would tell if two triangles were congruent. 3. If two triangles are congruent, what conclusions can you make? Give an example to illustrate your answer. Complete each congruence statement. 4. ARM 5. SPT Write a congruence statement for the congruent triangles in each diagram Testing for Congruent Triangles October 8, 001 8

9 Explain why the following pairs of triangles are not congruent. 8. Prove the following: 9. : AB RT AR AB BT RT AB RT AR TB Prove: ABR TRB 10. Refer to ALM and PRT. Name one additional pair of corresponding parts that need to be congruent in order to prove that ALM PRT. What postulate would you use to prove the triangles are congruent? Testing for Congruent Triangles October 8, 001 9

10 Determine whether each pair of triangles are congruent. If they are congruent, indicate the postulate that can be used to prove their congruence Write a two-column proof. : MO PQ NO bisects MP Prove: MNO PNO Testing for Congruent Triangles October 8,

11 Testing for Congruent Triangles Worksheet Key 1. Draw triangles TLA and RSB. Mark the corresponding parts for TLA RSB.. Describe how you would tell if two triangles were congruent. See if the six pairs of corresponding parts are congruent 3. If two triangles are congruent, what conclusions can you make? Give an example to illustrate your answer. The six pairs of corresponding parts are congruent. For example, if ABC RTS, then A R, B T, C S, AB RT, BC TS, and AC RS. Complete each congruence statement. 4. ARM LEG 5. SPT PSK Write a congruence statement for the congruent triangles in each diagram. 6. OAB AOD 7. MIT NIT Explain why the following pairs of triangles are not congruent. 8. The congruent parts are not corresponding. Testing for Congruent Triangles October 8,

12 Prove the following: 9. : AB RT AR AB BT RT AB RT AR TB Prove: ABR TRB Statement AB RT, AR AB, BT RT Reason AB RT, AR TB BAR and RTB are Right angles lines form four rt. angles BAR RTB All rt. Angles are ABR TRB If lines are cut by a transversal, alt. Interior angles are ABR TRB If angles in a are to angles in another, the third angles are also BR RB Congruence of segments is reflexive ABR TRB Definition of congruent triangles 10. Refer to ALM and PRT. Name one additional pair of corresponding parts that need to be congruent in order to prove that ALM PRT. What postulate would you use to prove the triangles are congruent? AM PR ASA Determine whether each pair of triangles are congruent. If they are congruent, indicate the postulate that can be used to prove their congruence. 11. ASA 1. SSS 13. Not congruent Testing for Congruent Triangles October 8, 001 1

13 14. Write a two-column proof. : MO PQ NO bisects MP Prove: MNO PNO Statement MO PQ Reason NO bisects MP MN PN Definition of bisector NO NO Congruence of segments is reflexive MNO PNO SSS Testing for Congruent Triangles October 8,

14 Name: Date: Class: Testing for Congruent Triangles Checklist 1. On questions 1 thru 3, did the student answer each question correctly? a. Yes (15 points) b. out of 3 (10 points) c. 1 out of 3 (5 points). On questions 4 and 5, did the student state the correct congruence statement? a. Yes (10 points) b. 1 out of (5 points) 3. On questions 6 and 7, did the student state the correct congruence statement? a. Yes (10 points) b. 1 out of (5 points) 4. On question 8, did the student explain why the triangles are not congruent? a. Yes (5 points) 5. On question 9, did the student write a correct proof? a. Yes (5 points) 6. On question 10, did the student name an additional pair of corresponding parts that need to be congruent in order to prove ALM TRB? a. Yes (5 points) 7. On question 10, did the student name the correct postulate to prove triangles congruent? a. Yes (5 points) 8. On questions 11 thru 13, did the student determine which pair of triangles are congruent? a. Yes (15 points) b. out of 3 (10 points) c. 1 out of 3 (5 points 9. On questions 11 thru 13, did the student indicate the correct postulate to prove the congruence? a. Yes (15 points) b. out of 3 (10 points) c. 1 out of 3 (5 points) 10. On question 14, did the student a correct write a two-column proof? a. Yes (5 points) Total Number of Points Testing for Congruent Triangles October 8,

15 NOTE: The sole purpose of this checklist is to aide the teacher in identifying students that need remediation. It is suggested that teacher s devise their own point range for determining grades. In addition, some students need remediation in specific areas. The following checklist provides a means for the teacher to access which areas need addressing. 1. Does the student need remediation in content (corresponding parts of congruent triangles) for questions 1 thru 3? Yes No. Does the student need remediation in content (completing congruence statements) for questions 4 and 5? Yes No 3. Does the student need remediation in content (writing congruence statements) for questions 6 and 7? Yes No 4. Does the student need remediation in content (explaining why triangles are not congruent) for question 8? Yes No 5. Does the student need remediation in content (proving a statement) for question 9? Yes No 6. Does the student need remediation in content (analyzing congruent triangles) for question 10? Yes No 7. Does the student need remediation in content (using postulates to prove congruence) for questions 11 thru 13? Yes No 8. Does the student need remediation in content (writing two-column proofs for congruence) for question 14? Yes No A 85 points and above B 81 points and above Sample Range of Points! C 7 points and above D 63 points and above F 6 points and below Testing for Congruent Triangles October 8,