UNIT 6 – C

92 downloads 606 Views 448KB Size Report
Chapter 4 Section 4: Triangle Congruence: SSS and SAS. I can determine if 2 ... ASSIGNMENT: Triangle Congruence and Logic Worksheet. Completed: Friday ...
Name _________________________________________

Period ____

11/2 – 11/13

GEOMETRY UNIT 6 – CONGRUENT TRIANGLES Vocabulary Terms: Congruent Corresponding Parts Congruency statement Included angle Included side

11/5

HL Non-included side Hypotenuse Leg

11/6 SSS and SAS

11/12

ASA, AAS, HL

11/7-8 Congruent Triangles and Logic

SSS SAS ASA AAS

11/2 Congruent Polygons 11/9 CPCTC

11/13 Review

Test

Friday, 11/2 Chapter 4 Section 3: Congruent Triangles I can match the corresponding parts of congruent figures given a picture or a congruency statement. I can prove polygons congruent using the definition of congruent polygons.

ASSIGNMENT: Pg. 234 (#2-11, 13-16, 19, 21-22, 28-31)

Completed:

Monday, 11/5 Chapter 4 Section 4: Triangle Congruence: SSS and SAS I can determine if 2 triangles are congruent using SSS or SAS.

ASSIGNMENT: Triangle Congruence WST - #2, 4, 8, 9, 13, 15, 16, 19

Completed:

Tuesday, 11/6 Chapter 4 Section 5: Triangle Congruence: ASA, AAS, and HL I can determine if 2 triangles are congruent using ASA, AAS, or HL.

ASSIGNMENT: Triangle Congruence WST - Finish

Completed:

Wednesday or Thursday, 11/7-8 Chapter 4 Section 4 and 5: Triangle Congruence: SSS and SAS AND ASA, AAS, and HL I can determine if 2 triangles are congruent using SSS, SAS, ASA, AAS, or HL. I can determine the missing piece of information needed to prove triangles congruent I can complete a fill-in-the-blank proof. I can use triangle congruence and logic to solve problems. ASSIGNMENT: Triangle Congruence and Logic Worksheet Completed:

Friday, 11/9 Chapter 4 Section 6: CPCTC I can use CPCTC to solve different types of problems. I can use CPCTC in geometric proofs

ASSIGNMENT: CPCTC Worksheet

Completed:

Monday, 11/12 Review Day Completed:

ASSIGNMENT: Review for Test Tuesday, 11/13

Test Day

Unit 6 Test: Congruent Triangles

Grade:

If you miss the review day, you are still expected to take the test on the test day. For more help BEFORE the test: 1. Use the indicated chapters in your book 2. Use the book online (it has videos and a homework help section) 3. Use Google to find more resources 4. Come to tutoring (with assignment)

CONGRUENT Polygons Examples

I. Name the congruent triangles. 1.

LIN ≅_______ I

2.

FOX ≅______

A O

L

N

R

X

E

B

II. Name the congruent triangle and the congruent parts.. 3.

E ≅  _____

FE ≅ _____

EFI ≅  _____

FI ≅ _____

FIE ≅  _____

IE ≅ _____

X

 FGH ≅ ______ EFI ≅  _____

FG ≅ _____

G ≅  _____

GH ≅ _____

H ≅  _____

FH ≅ _____

Use the congruency statement to fill in the corresponding congruent parts. 4.  EFI ≅ HGI

F

O

Example 3: Proving Triangles Congruent Given: ∠YWX and ∠YWZ are right angles. YW bisects ∠XYZ. W is the midpoint of XZ. XY ≅ YZ. Prove: ∆XYW ≅ ∆ZYW

Check It Out! Example 3 Given: AD bisects BE. BE bisects AD. AB ≅ DE, ∠A ≅ ∠D Prove: ∆ABC ≅ ∆DEC

Check It Out! Example 4 Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ≅ ML. JK || ML. Prove: ∆JKN ≅ ∆LMN

Name ________________________________________________ Period ______________ Triangle Congruence 1. List the five ways to prove that triangles are congruent. ______________ ______________ ______________ ______________ ______________ For each pair of triangles, tell which of the above postulates make the triangles congruent. 2. ∆ABC ≅ ∆EFD ______________

3. ∆AEC ≅ ∆BED

C

______________

C

B

F E

A

B

D

A

E

4. ∆ABC ≅ ∆EFD ______________

D

5. ∆ADC ≅ ∆BDC C

F

C

B

A

D

A

E

6. ∆ACE ≅ ∆DBE ______________

B

D

7. ∆ADC ≅ ∆BDC

B

C

______________

C

______________

E D

A

A

8. ∆ABC ≅ ∆CDA ______________ C

9. ∆ABE ≅ ∆CDE

B

B

D

______________

D

C

E

D

A

A

10. ∆CAE ≅ ∆DBE ______________ B

C

B

11. ∆MAD ≅ ∆MBC ______________ D

C

D E A

A

M

B

12. ∆DCA ≅ ∆DCB ______________

13. ∆ACB ≅ ∆ADB

C

______________

C

B

A

A

D

D

B

14. ∆RTQ ≅ ∆STP ______________

15. ∆DBA ≅ ∆BDC

______________

D

R

C

Q

T P

S

16. ∆AEB ≅ ∆DEC

A

______________

B

17. ∆CDE ≅ ∆ABF ______________

A

D

C F

E

C E

B

A

B

D

18. ∆DEA ≅ ∆BEC A

______________

19. ∆HIJ ≅ ∆QOP ______________ J

B E

H

I

Q

P

O D

C

20. ∆RTS ≅ ∆CAB T

______________

21. ∆ABC ≅ ∆ADC ______________ B

S C A

R A

C

B D

7. ∆BAP ≅ ∆BCP

______________

8. ∆SAT ≅ ∆SAR ______________ R

A

S

A B

D

P

T

C

Name ______________________________________________ Period _______________ Triangle Congruence and Logic Worksheet I. For each pair of triangles, tell: (a) Are they congruent (b) Write the triangle congruency statement. (c) Give the postulate that makes them congruent. 1.

D

2.

C

A

3.

B

B

A C

E

E

D

D A

C

B

a. ______________

a. ______________

a. ______________

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

c. ______________

c. ______________

c. ______________

5.

6.

4.

M

L

I

O

I S

W

E

S

H

E L

V

a. ______________

a. ______________

a. ______________

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

c. ______________

c. ______________

c. ______________

8.

9.

7.

L A

E

U

H

P A

T W

R

G

E

T M

a. ______________

a. ______________

a. ______________

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

b. ∆_____ ≅ ∆ _____

c. ______________

c. ______________

c. ______________

II. Using the given postulate, tell which parts of the pair of triangles should be shown congruent.

10. SAS

11. ASA

12. SSS

C

A

E D

B

F

F B

A

B A

E

D

C

C

D

_______ ≅ ________

13. AAS

________ ≅ ________

_______ ≅ _______

14. HL

15. ASA P

P

D

S

C

T R

A Q

_______ ≅ ________

R

B

Q

S

________ ≅ ________

_______ ≅ _______

III. Multiple Choice 16. Which set of coordinates represents the vertices of a triangle congruent to ∆RST? (Hint: Find the lengths of the sides of ∆RST) S T A. (3, 4) (3, 0) (0, 0) B. (3, 3) (0, 4) (0, 0) C. (3, 1) (3, 3) (4, 6) D. (3, 0) (4, 4) (0, 6) y

6

4

R 2

x 2

4

6

17. Given ∆ABC and ∆DEF. Which of the following pairs of corresponding parts would correctly prove the triangles congruent by ASA? A. ∠B ≅ ∠E , ∠A ≅ ∠D, AB ≅ DE B. ∠C ≅ ∠F , ∠A ≅ ∠D, AB ≅ DE C. ∠B ≅ ∠E , ∠C ≅ ∠F , AB ≅ DE

D. ∠B ≅ ∠E , ∠A ≅ ∠D, AC ≅ DF

For 18 – 19: Fill in the blank with the correct statement or reason to complete the proofs. 18. GIVEN: RZ bisects TS ; ∠3 ≅ ∠4 PROVE: ∆RZS ≅ ∆RZT

4 Z 3

2 1

R

T

S STATEMENTS

REASONS

1. RZ bisects TS 2. 3.

Definition of a segment bisector Given

4. RZ ≅ RZ 5. ∆RZS ≅ ∆RZT

A. Reflexive Property

B. Given

D. TZ ≅ ZS

E. SAS

C. ∠3 ≅ ∠4

S

P

19. GIVEN: ∠Q ≅ ∠S; R is the midpoint of QS . PROVE: ∆PRQ ≅ ∆TRS

R T

Q

STATEMENTS 1. ∠Q ≅ ∠S 2.

REASONS

Given

3. QR ≅ RS 4. 5. ∆PRQ ≅ ∆TRS

A. ∠PRQ ≅ ∠SRT D. Given 20.

Vertical Angle Theorem

B. Definition of midpoint

C. ASA

E. R is the midpoint of QS 21. Given: E is the midpoint of AB, C ≅ D Which of the following statements must be true?

A A ≅ D B AE ≅ ED C CE ≅ ED D CD ≅ BA

22. In the figure below, AC ≅ DF and C ≅ F Which additional information would be enough to prove ∆ABC ≅ ∆DEF?

A AB ≅ DE B AB ≅ BC C BC ≅ EF D BC ≅ DE

23.

24.

24. Draw a counterexample for the following statement. If the 3 angles in one triangle are congruent to the corresponding angles in another triangle, then the 2 triangles are congruent.

CPCTC Notes

I. CPCTC stands for ___________________________________________________________________________. This means that once you have proven the 2 triangles _______________________, you know that all the _________________________ parts are also ____________________.

Examples: 1) Find JK

2)

3) Additional Examples from worksheet: # 3, 5, and 16

CPCTC Worksheet I. Solve for the variable. 1. CAT ≅ DOG . Find h.

2.  IEF ≅ HGF . Find a.

A

E

F

G

O 15

h 38°



C I

DD

H

T G

3.  PQR ≅MNR . Find x.

4.  ABC ≅ ADC. Find y. A

Q

(3y)° x°

R

21°

M

P

35

D

o

B

C

N

II. Set up an equation and then solve for the variable. 5. CAT ≅ DOG . Find x.

6.  IEF ≅ HGF . Find a.

A

E

F

G

O (8x-6) (3x+9)

(5a) o

(2a+3) o

C I

DD

H

T G

7.  ABD ≅CDB. Find x.

8.  ABD ≅CDB. Find y.

A (2x+17) (4x-35) D

A

B

7y

B

o

o

C

D

3y + 20

C

III. For which value(s) of x are the triangles congruent?

9. x = _______________

10. x = _______________

D

D

5x - 8 F

E

A

3x + 2

5x° 92° E

C

A

B

11. x = _______________

A

B

m ∠3 = 3x m ∠4 = 7x - 10

3

E

7x - 4

4x + 8

2

4

C

13. x = _______________

14. x = _______________ C

D

19 B

D C

9x - 8

A

B

m ∠ CDB = (15x + 3)°

15. x = _______________ C W

(2x + 4)°

A

m ∠ ABD = (10x + 18)°

16. x = _______________

D

1

B

R

C

D

A

C

12. x = _______________

A

1

B

2

Z

x2 + 2x

x2 + 24

(4x – 6)° B

R

S

T

IV. Proofs 17. GIVEN: N is the midpoint of AB AX ≅ NY NX ≅ BY PROVE: ∠X ≅ ∠Y

X

Y

A

B

N REASONS

STATEMENTS 1. N is the midpoint of AB 2. 3. AX ≅ NY 4. 5.  AXN ≅ NYB 6.

Definition of a midpoint Given CPCTC T

18. GIVEN: RT ≅ RV TS ≅ VS PROVE: ∠RST ≅ ∠RSV

R

S

V REASONS

STATEMENTS 1. 2. 3. 4. 5. ∠RST ≅ ∠RSV

Given Reflexive SSS E

 19. GIVEN: VB bisects ∠EVO  BV bisects ∠EBO PROVE: ∠E ≅ ∠O

B

3 4

1 2 O REASONS

STATEMENTS

 1. VB bisects ∠EVO 2. 3.

Definition of Angle Bisector Given

4. ∠1 ≅ ∠2 5. BV ≅ BV 6. 7. ∠E ≅ ∠O

ASA

V