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°
a°
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