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Aim: How do we prove overlapping triangles are
congruent?
Do Now: State the names of two triangles in
each diagram:
A
D
1)
B
ABC,
B
2)
R
C
DCB
H
F
3)
S
Q
G
L
GSQ, LQS
GQL, LSG
FBH
MBR
M
4) A
G
P
K
D
V
APV , GKD
1
Proofs w/overlapping triangles:
1) Re-draw the diagram, separating the triangles.
2) Use Reflexive Postulate if any shared sides or angles
are needed in the proof.
Given:
A
D
A
D
AB  DC, AC  DB
Prove:
B
C B C
C
B
ABC  DCB
AB  DC, AC  DB
BC  BC
ABC  DCB
Given
Reflexive Postulate
S.S.S. Postulate
Geometry Lesson: Proofs With
Overlapping Triangles
2
Ex 1: Proof w/ overlapping triangles
R
Given: RE  TC, REP  TCG, PC  GE
Prove: REP  TCG
P
Statements
T
C
E
G
Reasons
1) RE  TC, REP  TCG
2) PC  GE
1) Given
3) CE  CE
3) Reflexive Postulate
4) PC  CE  GE  CE
5) PE  PC  CE , GC  GE  CE
4) Addition Postulate
6) PE  GC
7) REP  TCG
2) Given
5) Partition Postulate
6) Substitution Post.
7) S.A.S. Postulate
Geometry Lesson: Proofs With
Overlapping Triangles
3
Ex 2: Proof w/ overlapping triangles
Given: NQ  NT , QS and TV are medians
Prove: QS  TV
N
V
Q
Statements
QS  TV
T
Reasons
1) NQ  NT , QS and TV are medians
2) V is midpt. of NQ, S is midpt. of NT
3) NV  1 NQ, NS  1 NT
2
2
1
1
4)
NQ  NT
2
2
5)
NV  NS
N  N
6)
QNS  TNV
7)
8)
S
1) Given
2) Def. Median
3) Def. Midpoint
4) Mult. Postulate
5) Substitution Post.
6) Reflexive Post.
7) S.A.S. Postulate
Geometry Lesson: Proofs With
4
Overlapping Triangles 8) C.P.C.T.C
T
Proofs w/ overlapping triangles
1) Given: 1  4, 2  3, RD  SE
Prove: TDS  MER
P
2) Given: PF  PL, RF  SL
R
Prove: RFL  SLF , RL  SF
F
R
4
E
3
2
D
1
M
S
L
B
3) Given: BR  BH , RF  HM
Prove: RBM  HBF , F  M
R
F
Geometry Lesson: Proofs With
Overlapping Triangles
S
H
M
5