Testing for Congruent Triangles Examples

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.

2. 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.

4. ∠ 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.

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.

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? Î ∠ P is congruent to ∠ K

c. Which side of ∆ KLM is congruent to PSin ∆ PRS? Î KM is congruent to RS

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? Î ∠ 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.

Point out the importance of the order of the letters in a statement of congruence. ∆ PRS

≅ ∆ KLM, but ∆ PRS is not congruent ∆ MLK.

a. Prove that congruence of triangles is reflexive.

b. Prove that congruence of triangles is Symmetric.

c. Prove that congruence of triangles is transitive.

Given: ∆ XYZ

Prove : ∆ XYZ ≅ ∆ XYZ

Statements Reasons

∆ XYZ Given ∠ X ≅ ∠ X, ∠ Y ≅ ∠ Y, ∠ Z ≅ ∠ Z Congruence of ∠ ’s is reflexive.

XY ≅ XY , YZ ≅ YZ , XZ ≅ XZ Congruence of segments is reflexive.

∆ XYZ ≅ ∆ XYZ Definition of congruent triangles

Statements Reasons

∆ LMN ≅ ∆ OPQ Given

∠ L≅ ∠ O, ∠ M ≅ ∠ P, ∠ N ≅ ∠ Q, LM ≅ OP,

MN ≅ PQ ,LN ≅ OQ

CPCTC

∠ 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.

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,

DE

AB≅ , 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.

Given: ∆ ABC≅ ∆ DEF, ∆ DEF ≅ ∆ GHI Prove ∆ ABC≅ ∆ GHI

8.

9.

10. Example – Given ∆ ABC with vertices A(0, 5), B(2, 0), and C(0, 0) and ∆ RST with vertices R(5, 8), S(5, 3), T(3, 3), show that ∆ ACB ≅ ∆ RST.

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.

Use the distance formula to show that the corresponding sides are congruent. AC = (0−0)2 +(5−0)2 AC = 25 = 5 RS = (5−5)2 +(8−3)2 RS = 25 = 5 AB = (0−2)2+(5−0)2 AB = 29 RT = (5−3)2+(8−3)2 RT = 29 CB = (0−2)2+(0−0)2 CB = 4 = 2 ST = (5−3)2+(3−3)2 ST = 4 = 2

All the pairs of corresponding sides are congruent, so ∆ ACB ≅ ∆ RST by SSS. Given: AB ≅DB, AC ≅CD Prove: ABC DBC Statements Reasons DB AB≅ Given CD AC ≅ Given CB

BC ≅ Common Side (Reflexive Property)

11. Example – Given ∆PQR with vertices P(3, 4), Q(2, 2), and R(7, 2) and ∆STU with vertices S(6, -3), T(4, -2), and U(4, -7), show that ∆PQR≅ ∆STU.

12.

13. Example:

Use the distance formula to show that the corresponding sides are congruent. PQ = (3−2)2+(4−2)2 PQ = 5 PR = (3−7)2+(4−2)2 PR = 20= 2 5 QR = (2−7)2+(2−2)2 QR = 25= 5 TU = (4−4)2+(−2−(−7))2 TU = 25 = 5 ST = (6−4)2 =(−3−(−2))2 ST = 5 SU = (6−4)2+(−3−(−7))2 SU = 20= 2 5

All the pairs of corresponding sides are congruent, so ∆ PQR ≅ ∆ STU by SSS.

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.

Statements Reasons

YZ

WZ ≅ ,VZ ≅ZX Given

∠WZV ≅ ∠YZX Vertical ∠’s are ≅

∆ VZW ≅ ∆ XZY SAS

Given: WZ ≅YZ,VZ ≅ ZX

14. Example:

15. Example:

16.

Given: AC≅CD,BC≅CE

Prove: ABC DEC

Statements Reasons

CD

AC≅ Given

CE

BC ≅ Given

(Included Angle) ACB DCE Vertical Angles

ABC DEC SAS

Given: ∠1 and ∠2 are right angles, ST ≅TP

Prove: ∠3≅ ∠4

Statements Reasons

∠1 and ∠2 are right angles, ST ≅TP Given ∠1≅ ∠2 Al rt ∠’s are ≅ TR TR≅ Congruence of segments is reflexive PTR STR ≅∆ ∆ SAS ∠3≅ ∠4 CPCTC

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.

17. Example:

18. Example:

Given: ∠Q and ∠S are right angles, QR≅ SR Prove: ∠P≅ ∠T Statements Reasons SR QR ≅ Given ∠PRQ≅ ∠TRS Vertical ∠’s are ≅

∠Q and ∠S are right angles Given ∠Q ≅ ∠S All rt. ∠’s are ≅ ∆ PQR≅ ∆ TRS ASA ∠P ≅ ∠T CPCTC Given: BE bisects AD , ∠A≅ ∠D Prove: AB ≅DC Statements Reasons BE bisects AD , ∠A≅ ∠D Given ∠1≅ ∠2 Vertical ∠’s are ≅ ED AE ≅ Def. of bisector DEC AEB≅ ∆ ∆ ASA DC AB≅ CPCTC

Testing for Congruent Triangles Worksheet

1. Draw triangles TLA and RSB. Mark the corresponding parts for ∆TLA ≅ ∆RSB. 2. 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.

6.

7.

Name:___________________ Date:____________

Explain why the following pairs of triangles are not congruent.

8.

Prove the following:

9. Given:

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?

RT AB || AB AR⊥ RT BT ⊥ RT AB ≅ TB AR≅

Determine whether each pair of triangles are congruent. If they are congruent, indicate the postulate that can be used to prove their congruence.

11.

12.

13.

14. Write a two-column proof.

Given:

Prove: ∆MNO ≅∆PNO

PQ MO ≅

Testing for Congruent Triangles Worksheet Key

1. Draw triangles TLA and RSB. Mark the corresponding parts for ∆TLA

≅ ∆RSB.

2. Describe how you would tell if two triangles were congruent. ÎSee 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. ÎThe 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.

7.

Explain why the following pairs of triangles are not congruent.

8. The congruent parts are not corresponding. AOD OAB≅∆ ∆ NIT MIT ≅∆ ∆

Prove the following:

9. Given:

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?

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 12. SSS 13. Not congruent RT AB || AB AR⊥ RT BT ⊥ RT AB≅ TB AR≅ Statement Reason RT AB || , AR⊥ AB, BT ⊥RT RT AB ≅ , AR≅TB Given

∠BAR and ∠RTB are Right angles ⊥ lines form four rt. angles

∠BAR

≅

∠ABR

≅

≅

∠ABR

≅

≅

≅

RB

BR≅ Congruence of segments is reflexive

∆ABR ≅∆TRB Definition of congruent triangles

PR AM ≅

14.Write a two-column proof.

Given:

Prove: ∆MNO ≅∆PNO

PQ MO ≅ NO bisects MP Statement Reason PQ MO≅ NO bisects MP Given PN MN ≡ Definition of bisector NO

NO ≅ Congruence of segments is reflexive

PNO

MNO≅∆

Testing for Congruent Triangles Checklist

1. On questions 1 thru 3, did the student answer each question correctly?

a. Yes (15 points) b. 2 out of 3 (10 points)

c. 1 out of 3 (5 points)

2. On questions 4 and 5, did the student state the correct congruence statement?

a. Yes (10 points) b. 1 out of 2 (5 points)

3. On questions 6 and 7, did the student state the correct congruence statement?

a. Yes (10 points) b. 1 out of 2 (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. 2 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. 2 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 _________ Name: ____________

Date:__________ Class:_____________

1. Does the student need remediation in content (corresponding parts of congruent triangles) for questions 1 thru 3? Yes__________ No__________

2. 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

C 72 points and above

D 63 points and above

F 62 points and below

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.

Sample Range of Points!