Download Corresponding angles and corresponding sides are congruent in

Unit 4: Congruence and Triangles (4.2) p.202­211
Congruent Triangles:2 triangles that are exactly the same, both equal in length and equal in angle measure
Corresponding angles and corresponding sides are congruent in congruent triangles. see example below
Ex: Triangle ABC ≅ Triangle DEF
E
B
D
A
C
F
1
Third Angles Theorem: If 2 angles are congruent in one triangle to 2 angles in another triangle, then the third angle in each triangle will also be congruent
B
D
F
A
C
E
If <A≅<D and <B≅<E, then <C≅<E
2
Ex: Solve for x and y
I
ABCDE≅FGHIJ
A
111o
E 93o
B
15 in.
14 in.
105o
D 12 in. C
(4x + 7) in.
(2y­7)o J
H
G
F
3
Ex: Solve for x
(5x + 8)o
42o
(4x ­ 1)o
4
Properties of Congruent Triangles
Reflexive Property of Congruent Triangles: a triangle is congruent to itself
ABC ≅ ABC
A
Symmetric Property of Congruent Triangles
: DEF ≅ ABC
If ABC ≅ DEF, then D
Transitive Property of Congruent Triangles
:
If and , ABC ≅ DEF
DEF ≅ JKL
then ABC ≅ JKL
J
B
C
E
F
K
L
5
X
Ex: Given: XY // WZ
XY ≅ WZ
V is the midpoint of XZ and WY
Prove: YVX ≅ WVZ
W
V
Y
Z
StatementsReasons
1) XY // WZ, XY ≅ WZ
1) _________________
2) <YXV≅<VZW
<XYV≅<ZWV
2) _________________
3)__________________
3) Vertical <'s Theorem
4) V is the mdpt XZ & WY 4)__________________
5)___________________
5) __________________
6) YVX ≅ WVZ
6) __________________
6
HW: p.206­207
#10­14
#16­21
#23­28
7
Unit 4: Proving Triangles are Congruent: SSS & SAS(4.3)p.212­219
5 Congruence Postulates and Theorems: THE BIG 5
1: Side Side Side Postulate (SSS)
2: Side Angle Side Postulate (SAS)
3: Angle Side Angle Postulate (ASA)
4: Angle Angle Side Postulate (AAS)
5: Hypotenuse Leg Theorem (HL)
8
Side­Side­Side (SSS) Postulate: SSS: If 3 sides of one triangle are congruent to the 3 corresponding sides of another triangle, then the two triangles are congruent.
D
B
A
C
F
E
9
Side­Angle­Side (SAS) Postulate: SAS: If 2 sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then two triangles are congruent.
B
M
N
A
C
L
10
Ex: Given AB≅DB and AC≅DC
Prove: ABC≅ DBC
A
Statements
C
B
D
Reasons
1)
1) Given
2)
2)
3)
3)
11
Ex: Given: C is the midpoint of AE and BD
B
Prove: ABC ≅ EDC E
C
A
Statements
D
Reasons
1)
1)
2)
2)
3)
3)
4)
4)
12
T
Given: UV bisects TW at point V
TU≅WU
Prove: TVU ≅ WVU
V
U
Statements W
Reasons
1)
1) Given
2) V is the midpoint of 2)
TW
3)
3) Def. of midpoint
4)
4) Given
5)
5)
6)
6)
13
Ex: Is there enough information to prove the 2 triangles congruent? If so, give a reason why.
and the congruence statement
C
a)
E
F
A
B
D
N
Z
b)
X
Y
M
L
T
c)
V
R
Q
R
d) N
M
L
Q
14
HW: p.216­217
#11­23 all
15
Unit 4: Proving Triangles are Congruent: ASA & AAS (4.4) p.220­228
ASA: If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle, then the 2 triangles are congruent.
Z
A
B
C
X
Y
16
AAS: If 2 angles and a non­included side of one triangle are congruent to 2 angles and a non­included side of another triangle, then the 2 triangles are congruent.
A
H
C
B
K
J
17
HL: If the hypotenuse and a leg of one right triangle are congruent to a hypotenuse and a leg of another right triangle, then the 2 triangles are congruent.
B
A
C
I
G
H
18
Given: <A≅<C and <ADB and <BDC are right angles
Prove: ADB ≅ CDB
A
Statements
1)
2) <ADB≅<BDC
B
D
C
Reasons
1)
2)
3)
3)
4)
4)
19
Given: AC≅CD and <A≅<D
Prove: ABC ≅ DEC
E
B
C
A
Statements
D
Reasons
1)
1)
2)
2)
3)
3)
20
A
Given: AZ≅AX <AYX and <AYZ are right angles
Prove: AYZ ≅ AYX
Z
Reasons
Statements
1)
1)
2)
2)
3)
3)
4)
4)
Y
X
21
E
Given: BE bisects <AED
<A≅<D
Prove: ABE ≅ DBE
A
Statements
B
D
Reasons
1) BE bisects <AED
1)
2) 2)
3) <A≅<D
3)
4)
4)
5)
5)
22
Ex: Is there enough information to prove the triangles are congruent? If so, explain why?
D
B
a)
C
E
A
Y
b) Z
X
W
O
c) M
N
P
L
d) T
S
Q
R
23
HW: p.223­224
#8­22 all
24
Review Congruent Triangles­­Extra Practice with THE BIG FIVE: SSS, SAS, ASA, AAS, HL
Example 1: Are the 2 triangles congruent? If so, why are they congruent and give the congruence statement.
B
b) A
R
Z
a)
C
X
Y
T
Q
E
D
25
Ex 1: continued
c)
H
e)
R
E
N
d)
X
T
S
F
G
A
O
f)
M
L
O
C
M
E
Y
N
26
Example 1: continued
S
D
g)
A
h)
U
C
B
Q
T
i)
R
j)
T
D
G
E
F
U
X
W
V
27
Unit 4: Using Congruent Triangles (4.5) p.229­242
TIDBITS OF INFO
­There is no Angle Side Side when proving congruent triangles ­Once you use 1 of THE BIG 5, the reason you can use to say all other parts of triangles are congruent is:
Corresponding Parts of Congruent Triangles are Congruent (CPCTC): If 2 triangles are congruent, then all corresponding parts (sides and angles) are also congruent.
­Vertical angles are always congruent
E
A
<ACB ≅ <ECD
C
D
B
­Reflexive sides are always congruent
N
LN ≅ LN
M
P
L
­Parallel Lines only allow you to say 2 angles are congruent
A
E
<A≅<D
<E≅<B
C
B
D
A
B
C
<ABC≅<E
E
D
28
B
Given: AB≅BC and AD≅AC Prove: <A≅<C
A
Reasons
Statements
1)
1)
2)
2)
3)
3)
4)
4)
D
C
29
Given: AB//DC and BD//AC
Prove: AB≅CD
B
A
Statements
1)
2) <ABC≅______
<ACB≅______
D
C
Reasons
1)
2)
3)
3)
4)
4)
5)
5)
30
Given: QR//VT , QR≅VT, and
<R≅<T
Prove: QV≅VX
Q
R
V
T
X
Statements 1) QR//VT
Reasons
1)
2)
2)
3)
3) Given
4)
4)
5)
5)
31
Given: <WVY≅<WXZ and WZ≅WY
Prove: <Z≅<Y
V
Z
Reasons
Statements
1)
1)
2)
2)
3)
3)
4)
4)
W
X
Y
32
HW: p.232­234
#8­18 all
33