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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