Triangle Congruence Theorems (5.3, 5.5, 5.6, 5.7)
A
Name the sides of Angle C: _______________________
Angle C would be the included angle for those two sides!
Name the included angle for sides BA and AC: ____________________
B
C
Is A the included angle for sides BC and CA? ____________________
What do you think the included SIDE would be for Angles A and B? ______________
What is the included side for Angles B and C? __________________
Side AC is the included side for what two angles? ______________________
Congruent triangles have _______________________ and ________________________
If the two triangles satisfy the conditions of these 5 theorems, then they are congruent
The 5 Triangle Congruence Theorems:
1. SSS- Side, Side, Side:
2. SAS- Side, Angle, Side: (Included Angle)
3. ASA- Angle, Side, Angle: (Included Side)
4. AAS- Angle, Angle, Side: (Non-Included Side
5. HL- Hypotenuse, Leg:
NOTE: There is NO SUCH THING as Angle, Side, Side! It must be the included angle for the
two triangles to be congruent!
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Section 5.3- Triangle Congruence – SAS (Side-Angle-Side)
Decide whether enough information is given to prove that the triangles are congruent using the
SAS Congruence Theorem
Can you use the SAS Congruence Theorem? If so, finish the congruence statement.
Yes/ No ____________
Yes/ No ____________
∆𝐴𝐵𝐶 ≅ ∆______________
∆𝐴𝐵𝐷 ≅ ∆______________
Yes/ No ____________
Yes/ No ____________
∆𝐴𝐶𝐵 ≅ ∆______________
∆𝐿𝑀𝑁 ≅ ∆______________
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Section 5.5- Congruence – SSS (Side-Side-Side) – HL (Hypotenuse-Leg)
Decide whether enough information is given to prove that the triangles are congruent using the
SSS or HL Congruence Theorem
Can you use the SSS or HL Congruence Theorem? If so, finish the congruence statement
Yes: ___________
or
No
or
∆𝐴𝐵𝐷 ≅ ∆______________
or
No
∆𝐴𝐵𝐶 ≅ ∆______________
∆𝐽𝐾𝐿 ≅ ∆______________
Yes: ___________
Yes: ___________
No
Yes: ___________
or
No
∆𝑃𝑄𝑆 ≅ ∆______________
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Section 5.6- ASA (Angle-Side-Angle) – AAS (Angle-Angle-Side)
Decide whether enough info is given to prove that the triangles are congruent by ASA or AAs
Can you use the ASA or AAS Congruence Theorem? If so, finish the congruence statement
Yes: ___________
or
No
∆𝑃𝑄𝑆 ≅ ∆______________
Yes: ___________
or
∆𝐴𝐵𝐶 ≅ ∆______________
Yes: ___________
or
No
∆𝐴𝐵𝐷 ≅ ∆______________
No
Yes: ___________
or
No
∆𝐼𝐾𝑀 ≅ ∆______________
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Congruent Triangles
Name____________________
Tell whether the SSS, ASA,AAS,SAS or HL postulates can be applied to prove the triangles congruent. Write a
congruence statement. Don’t forget what you may assume from a diagram. If the triangles cannot be proved
congruent, write not possible and do not include a congruence statement.
1.
A
E
B
2.
J
N
3.
I
L
G
K
H
C
4. O
K
D
F
N
R
5.
T
S
U
6.
M
Z
R
X
Y
W
P
What additional information would you need to prove the triangles congruent by HL?
7.
E
B
8. In FGH and IJK, FG  IJ and FH  IK .
A
D
C
Draw and label a figure to show the congruent triangles.
9. If LMN  OPR, mL = 29, mP = 66, mN = (4x + 53). Find x and mR.
10. If LMN  OPR, LM is 10 less than 3 times a number, LN is 2 less than twice the number, PR is 5 more
than the number, OP is 4 more than the number, find LN and OR
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Determine what additional information is needed to enable you to use the indicated method
to prove that  ABC   DEF?
1.
 A   D, AC  DF ; ASA Congruence
2.
 C and  F are right angles, AC  DF ; HL Congruence
3.
 E   B, AB  DE ; SAS Congruence
Draw a picture for each problem and then answer each question.
4.
In isosceles  ABC where AB  CB and D is midpoint of AC , then  ABD   CBD by:
a.
5.
HL
b.
SSS
c.
SAS
d.
either SSS or SAS or HL
In  ABC and  RST, m A = 82°, m S = 76°, m C = 22° ,  A   R and AC  RT .
The two triangles are congruent by:
a.
6.
7.
8.
AAA
b.
ASA or AAS
c.
SSA
d.
not enough information
To prove the two triangles congruent by HL, what additional information must be known?
A
a.
AR
c.
AB  BC
b.
T  A
d.
 ABC and  RST are right angles
B
R
C
S
A
T
If  ABC  DEF and  EDF  RST, then what conclusion follows?
a.
 ABC  RST
c.
 ABC  SRT
b.
 DEF  RST
d.
 ABC  TRS
In  ABC, AB  CB , m A = 2x  10 and m B = 3x + 25.
Find m B.
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Examples Page:
EDC
EDC
98
1.
3.
4.
Vertical Angles
99
A
Triangle Congruence Proofs
Given:
AB ∥ CD
AB ≅ CD
Prove:
Triangle ∆ ABE ≅ ∆ DCE
B
E
C
STATEMENTS
D
DREASONS
G
Given: B is the midpoint of AR
GB ⊥ AR
A
Prove: AG ≅ RG
STATEMENTS
B
R
REASONS
B is the midpoint of AR
GB ⊥ AR
AG ≅ RG
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Solving Proofs with Congruent Triangles:





List all given facts and mark figures accordingly
Tell what given info means (Def of…, 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑛 ∥ 𝑙𝑖𝑛𝑒𝑠)
Mark all vertical angles and shared sides
Write a triangle congruence statement and the reason
Use CPCTC to show congruency of corresponding parts
Give some examples of a statements and reasons when you would use “Definition of”
Statement
Drawing
A
Reason
C
B
Definition of ________________
Definition of ________________
𝐺𝑅 ⊥ 𝐴𝐵
G
Definition of ________________
A B R
Definition of ________________
Whenever parallel lines are present, what are three types of angles that often occur?
Statements
Drawing
Reason
101
102
Given the congruency statement, list 6 different pairs of corresponding parts that could be listed with the reason
CPCTC:Δ𝐴𝐵𝐶 ≅ ∆𝑅𝑊𝑌
Ex. 3
Ex 4.
VTU
EDC
103
1.
2.
EDC
3.
4.
5.
6.
Corresponding
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Name: _________________________ Date: _____________ Hour: ____________
Triangle Congruence Review
Short Answer/Vocabulary
1) Name the five Triangle Congruence Properties
2) Give an example of a Congruence Statement
3) What does CPCTC stand for?
4) What is CPCTC used to prove?
Use the following congruence statements to find segments or angles that are congruent to what is given:
∆ABC ≅ ∆DEF
5) AB≅ _______
6) BC≅ _______
7) ∠A≅ _______
8) ∠F≅ _______
9) If ∆ABC ≅ ∆DEF and ∆EFD ≅ ∆RST, which of the following congruence statements is also true?
A) ∆ABC ≅ ∆RST
B) ∆ABC ≅ ∆EDF
C) ∆ABC ≅ ∆TRS
D) ∆ABC ≅ ∆SRT
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State if the triangles are congruent by SSS, SAS, ASA or AAS and HL. Then give a congruency statement for
each. If they are not congruent, state not congruent, and do NOT include a congruency statement:
10)
11)
J
I
A
B
S
12)
E
G
K
H
C
13)
X
D
T
∆ABC ≅ ∆ ________
∆IJK ≅ ∆ ________
Z
F
14)
R
O
W
U
∆STU ≅ ∆ ________
15)
N
R
N
L
Y
K
M
P
∆XYZ ≅ ∆ ________
∆OPR ≅ ∆_________
∆KLN ≅ ∆ ________
For each of the following problems, draw and label a figure to show the congruent triangles, then solve
the problem:
16) If  CAT   DOG, CA = 14, AT = 18, TC= 21 and DG = 2x + 7, find the value of x.
17) If  JKL   ABC, m J = 37°, m B = 64°, and m C = (3x + 52)°, find the value of x.
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18) If  PQR   CDE, PQ is 10 less than 3 times a number, PR is 2 less than twice the number, DE is 5 more
than the number, CD is 4 more than the number, find PQ and CE.
Determine what additional information is needed to enable you to use the indicated method to prove that
 ABC   DEF?
19)  A   D, AC  DF ; ASA Congruence
20)  C and  F are right angles, AC  DF ; HL Congruence
21)  E   B, AB  DE ; SAS Congruence
A
Triangle Congruence Proof:
B
22) Given: AB ∥ CD
∠C ≅ ∠B
Prove:
AC ≅ BD
STATEMENT:
C
D
REASON:
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