Download Section 4.2

Section 4.2
IDENTIFYING CONGRUENT FIGURES
PROVING TRIANGLES ARE CONGRUENT
IDENTIFYING CONGRUENT FIGURES
Congruent Polygons
• Are the same size and the same shape.
– Fit exactly on top of each other
• Have  corresponding parts:
– Matching sides and s
Identifying Congruent Figures
• Two geometric figures are congruent if they
have exactly the same size and shape.
NOT CONGRUENT
CONGRUENT
You can make 3 kinds of moves so
that one congruent figure can fit
exactly on top of top of another
Triangles
Corresponding angles
A ≅ P
B ≅ Q
C ≅ R
Corresponding Sides
AB ≅ PQ
BC ≅ QR
CA ≅ RP
B
A
Q
CP
R
Naming Polygons
• Order Matters!!
A
B
C
W
U
T
AB  WU
ED  RS
E
D
R
ABCDE  WUTSR
S
B  U
D  S
A  W
Example: ΔWYS  ΔMKV
• mW  mM = 25
• mY  mK = 55
• Find mV S=100
K
Y
55
W
25
S
M
100
V
Example: Congruence Statement
Finish the following congruence statement:
ΔJKL  Δ_ _ _
M
J
L
K
N
ΔJKL 
ΔNML
Example Naming congruent parts
The two triangles are congruent. Write a congruence statement.
Identify all parts of congruent corresponding parts.
• Angles:
D≅ R
E ≅ S,
F ≅T
• Sides
DE ≅ RS,
EF ≅ ST,
FD ≅ TR
• ∆ DEF ≅ ∆ RST.
F
R
E
S
D
T
Example Using properties of congruent figures
• In the diagram NPLM ≅ EFGH
• Find the value of x. • You know that LM ≅ GH.
• So, LM = GH.
8 = 2x – 3
11 = 2x
11/2 = x
F
8m
L
G
110°
87°
P
M
(7y+9)°
72°
10 m
N
(2x - 3) m
E
H
Example Using properties of congruent figures
• In the diagram NPLM ≅ EFGH
• Find the value of y
• You know that N ≅E.
So, mN = mE.
72°= (7y + 9)°
63 = 7y
9=y
F
8m
L
M
G
110°
72°
87°
P
(7y+9)°
10 m
N
(2x - 3) m
E
H
Third Angles Theorem
• If any two angles of one triangle are congruent
to two angles of another triangle, then the third
angles are also congruent.
• If A ≅ D and B ≅ E, then C ≅ F.
B
C
A
E
D
F
Proof: Third Angles Theorem
• If 2 s of one Δ are  to 2 s another Δ, then
the third s are also .
• Given: B  E
A  D
• Prove: C  F
A
B
C
D
E
F
Statements
Reasons
1) B  E & A  D
1) Given
2) mB + mA + mC = 180
2) Def. of Δ
mE + mD + mF = 180
3) mB + mA + mC =
3) Trans.
mE + mD + mF
4) mB + mA + mC =
4) Subs.
mB + mA + mF
5) mC = mF
5) Subtr.
6) C  F
6) Def. of 
Example Using the Third Angles Theorem
• In the diagram, N ≅ R and
L ≅ S.
• From the Third Angles
Theorem, you know that M
(2x + 30)° T
≅ T.
• So mM = mT. From the
Triangle Sum Theorem,
mM=180° - 55° - 65° = 60°
• mM = mT
60° = (2x + 30)°
S
30 = 2x
15 = x
• Find the value of x.
M
R
55°
N
65°
L
PROVING TRIANGLES ARE CONGRUENT
THEOREM
Properties of Congruent Triangles
B
Reflexive Property.
Every triangle is congruent to itself.
C
E
A
F
D
Symmetric Property.
If ABC  DEF , then DEF  ABC
Transitive Property.
K
L
If ABC  DEF and DEF  JKL, then ABC  JKL
J