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Proving Δ’s are  using:
SSS, SAS, HL, ASA, & AAS
HOMEWORK:
WS - Congruent Triangles
Methods of Proving Triangles Congruent
SSS
SAS
ASA
AAS
HL
If three sides of one triangle are congruent to three sides of
another triangle, the triangles are congruent.
If two sides and the included angle of one triangle are congruent
to the corresponding parts of another triangle, the triangles are
congruent.
If two angles and the included side of one triangle are congruent
to the corresponding parts of another triangle, the triangles are
congruent.
If two angles and the non-included side of one triangle are
congruent to the corresponding parts of another triangle, the
triangles are congruent.
If the hypotenuse and leg of one right triangle are congruent to
the corresponding parts of another right triangle, the right
triangles are congruent.
DIRECT Information
Direct information comes in two forms:
congruent statements in the ‘GIVEN:’ part of a proof
marked in the picture
Example:
GIVEN
PROVE
KL
NL, KM
KLM
NM
NLM
OR
INDIRECT Information
Indirect Information appears in the ‘GIVEN:’ part of the
proof but is NOT a congruency statement
Example:
J
Given: JO  SH;
O is the midpoint of SH
Prove:  SOJ  HOJ
S
O
H
INDIRECT Information
• Perpendicular lines  right angles  all rt ∠s are ≅
• Midpoint of a segment  2 ≅ segments
• Parallel lines  AIA
• Parallelogram  2 sets of parallel lines  2 pairs of AIA
• Segment is an angle bisector  2 ≅ angles
• Segments bisect each other  2 sets of ≅ segments
• Perpendicular bisector of a segment  2 ≅ segments &
2 right angles
BUILT-IN Information
Built- in information is part of the drawing.
Example:
Vertical angles  VA
Shared side  Reflexive Property
Shared angle  Reflexive Property
Any Parallelogram 2 pairs parallel lines  2 pairs of AIA
Steps to Write a Proof
1. Take the 1st Given and MARK it on the picture
2. WRITE this Given in the PROOF & its reason
3. If the Given is NOT a ≅ statement,
write the ≅ stmt to match the marks
Continue until there are no more GIVEN
4. Do you have 3 ≅ statements?
If not, look for BUILT-IN parts
5. Do you have ≅ triangles?
If not, write CNBD
If YES, Write the triangle congruency
and reason (SSS, SAS, SAA, ASA, HL)
GIVEN
KL ≅ NL, KM ≅ NM
PROVE
𝐾𝐿  𝑁𝐿
KLM ≅
NLM
given
𝐾𝑀  𝑁𝑀
given
𝐿𝑀  𝐿𝑀
reflexive prop
ΔKLM ≅ ΔNLM SSS
GIVEN BC ≅ DA, BC AD
PROVE ΔABC ≅ ΔCDA
BC ≅ DA
BC AD
given
given
∠BCA ≅ ∠DAC
AC ≅ AC
AIA
reflexive prop
ΔABC ≅ ΔCDA
SAS
Given: A  D, C  F, 𝐵𝐶  𝐸𝐹
Prove: ∆ABC  ∆DEF
A
D
B
F
A  D
given
C  F
given
𝐵𝐶  𝐸𝐹
given
∆ABC  ∆DEF
C
E
AAS
Given: 𝐿𝐽 bisects IJK,
ILJ   JLK
Prove: ΔILJ  ΔKLJ
𝐿𝐽 bisects IJK
IJL  IJH
ILJ   JLK
𝐽𝐿  𝐽𝐿
I
J
L
K
Given
Definition of angle bisector
Given
Reflexive Prop
ΔILJ  ΔKLJ
ASA
Given: 𝑇𝑊 ≅ 𝑉𝑊, 𝑈𝑉 ≅ 𝑉𝑋
Prove: ΔTUV  ΔWXV
U
W
𝑇𝑊 ≅ 𝑉𝑊
𝑈𝑉 ≅ 𝑉𝑋
TVU  WVX
Given T
Given
Vertical angles
 ΔTUV  ΔWXV
SAS
V
X
Given: 𝐻𝐽 ≅ 𝐽𝐿, H L
Prove: ΔHIJ  ΔLKJ
I
𝐻𝐽 ≅ 𝐽𝐿
H L
IJH  KJL
K
Given
Given
Vertical angles
J
H
 ΔHIJ  ΔLKJ ASA
L
Given: 𝑃𝑅 ≅ 𝑆𝑇, PRT  STR
Prove: ΔPRT  ΔSTR
𝑃𝑅 ≅ 𝑆𝑇
PRT  STR
𝑅𝑇 ≅ 𝑅𝑇
R
S
Given
Given
Reflexive Prop
ΔPRT  ΔSTR SAS
P
T
Given: 𝑀𝐵 is perpendicular bisector of 𝐴𝑃
Prove: ∆𝐴𝐵𝑀 ≅ ∆𝑃𝐵𝑀
𝑀𝐵 is perpendicular bisector of 𝐴𝑃
given
∠ABM & ∠PBM are rt ∠s
def  lines
∠ABM ≅ ∠PBM
all rt ∠s are ≅
𝐴𝐵 ≅ 𝐵𝑃
𝐵𝑀 ≅ 𝐵𝑀
def  bisector
reflexive prop.
ΔABM ≅ ΔPBM
SAS
Given: O is the midpoint of 𝑀𝑄 and 𝑁𝑃
Prove: ΔMON ≅ ΔPOQ
O is the midpoint of 𝑀𝑄 and 𝑁𝑃
given
𝑀𝑂 ≅ 𝑂𝑄
def. midpoint
𝑁𝑂 ≅ 𝑂𝑃
def. midpoint
∠MON ≅ ∠QOP
VA
ΔMON ≅ ΔQOP
SAS
Given: 𝐴𝐷 ≅ 𝐶𝐷; 𝐴𝐷 || 𝐶𝐷
Prove: ΔABD ≅ ΔCDB
𝐴𝐷 ≅ 𝐶𝐷
given
𝐴𝐷 || 𝐶𝐷
given
∠ADB ≅ ∠CBD
AIA
𝐵𝐷 ≅ 𝐵𝐷
reflexive prop.
ΔABD ≅ ΔCDB
SAS
J
Given: 𝐽𝑂  𝑆𝐻;
O is the midpoint of 𝑆𝐻
Prove:  SOJ  HOJ
S
0
H
Given: HJ  GI, GJ  JI
Prove: ΔGHJ  ΔIHJ
H
G
J
I
Given: 1  2; A  E ; C is midpt of AE
Prove: ΔABC  ΔEDC
B
D
1
A
2
C
E
Given: 𝑃𝑄  𝑄𝑅 , 𝑃𝑆  𝑆𝑅 , and 𝑄𝑅  𝑆𝑅
Prove: ΔPQR  ΔPSR
𝑃𝑄  𝑄𝑅
PQR = 90°
𝑃𝑆  𝑆𝑅
PSR = 90°
PQR  PSR
𝑄𝑅  𝑆𝑅
𝑃𝑅  𝑃𝑅
Given
P
Def.  lines
Given
Def.  lines
all right s are 
Given
Reflexive Prop
ΔPQR  ΔPSR
HL
Q
R
S
Checkpoint
Decide if enough information is given to prove the
triangles are congruent. If so, state the
congruence postulate you would use.
Given: LJ bisects IJK, ILJ   JLK
Prove: ΔILJ  ΔKLJ
I
J
L
K
Given: 1  2, A  E and 𝐴𝐶  𝐸𝐶
B
Prove: ΔABC  ΔEDC
1  2
A  E
𝐴𝐶  𝐸𝐶
Given
Given
Given
ΔABC  ΔEDC
D
1
A
ASA
2
C
E
Given: 𝐴𝐵  𝐶𝐵 , 𝐴𝐷  𝐶𝐷
Prove: ΔABD  ΔCBD
𝐴𝐵  𝐶𝐵
𝐴𝐷  𝐶𝐷
𝐵𝐷  𝐵𝐷
A
Given
Given
Reflexive Prop
 ΔABD  ΔCBD
C
B
SSS
D