Download Chap 4 Notes

-1-
Name:_______________________________
Geometry Rules!
Chapter 4 Notes
Period:______
Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles)
Congruent Figures
Corresponding Sides
Corresponding Angles
Triangle Angle-Sum Theorem
If two ________ of one triangle are _____________ to two _________ in another
triangle then the third angles in both triangles are _____________.
Practice:
1.) If ∆CAT  ∆DOG, then complete: (draw a picture first)
mC  _____
TCA  _____
GD  _____
O  _____
TA = _____
ODG  _____
2.) ZAK  JOE
a) Name three pairs of corresponding angles:
b) Name three pairs of corresponding sides:
-1-
-2-
3.) The two triangles shown are congruent; complete. (It will help to rotate the triangles first,
to get them in corresponding positions)
a) RAV  _____
b) R  _____
c) EV = _____
d) mA  _____
e) NV  _____
f) VRA  _______
E
N
V
R
A
Isosceles Triangles
Isosceles Triangle Theorem (
)
If two sides of a triangle are congruent, then the angles opposite them are ___________.
Converse of the Isosceles Triangle Theorem (
)
If two angles of a triangle are congruent, then the __________ opposite them are _________.
-2-
-3-
Angle bisector in vertex angle of Isosceles Triangle:
Equilateral Triangles
Practice: Solve for x and y
4.)
5.)
x
x
7y - 5
3y + 7
40
y
6.)
7.)
64
x
y
3x - 2
9
y
58
100
12
5x - 10
40
8.) In equilateral ∆XYZ, mX  a  b
and mY  2a  b . Find a and b.
9.) In equiangular ∆ABC, AB = 2x + y,
BC = 6x – 2y, and AC = 10. Solve for
x and y.
-3-
-4-
10.) What can you conclude from the picture?
F
10 cm
A
10 cm
E
10 cm
G
10 cm
C
B
B
11.)
Given:
C is the midpoint of BD
1  2
Prove:
A
2
1
AB  CD
D
Reasons
Statements
1.)
1.)
2.)
2.) Definition of Midpoint
3.)
C
AB  BC
3.)
4.)
4.)
12.)
B
Given: 1  4
Prove: AB  BC
A
1
2
3
C
4
Reasons
Statements
1.)
1.)
2.)
2.)
3.)
3.) Substitution
4.)
4.)
-4-
-5-
Algebra Review: Collecting like terms
Simplfy:
1.) x 2  3x  2 x 2  x
2.) 2 y 2  6 y  3 y 2  4 y  9  y  5
3.) 3x 2  8 x  4 x 2  2 x  5  x 2
4.)
5.) x2 y  3xy  5x  4 y  5xy 2  3x 2 y
6.) 2 xy  3xy 2  x2  5 y  4 xy  xy 2
7.) 2 x 2  5 x  15  2 x  2 x 2  9
8.) 12 x 2  3x  5  3x 2  6  3x  7 x 2
16 y 2  5 y  4  7 y  5 y 2  10
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Notes #23: Sections 4.2 and 4.3 (Methods of Proving Triangles Congruent)
Q: How can we prove that two triangles are congruent to each other?
A: Four ways: SSS, SAS, ASA, AAS
SSS:
_______-________-________ Postulate
SAS:
_______-________-________ Postulate
ASA:
_______-________-________ Postulate
AAS:
_______-________-________ Postulate
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-7-
Are the triangles congruent? If so, write the congruence and name the postulate used.
 Redraw your triangles so they line up
 You need three congruent pairs of sides/angles to follow:
SSS, SAS, ASA, or AAS
 Look for “hidden” pieces in:
- vertical angles
- overlapping sides
- congruent angles formed by parallel lines
- bisected angles
- ITT/Converse of ITT
- midpoints
1.)
2.)
V
Q
R
P
O
W
X
U
S
T
 ______   ______ by ________
 ______   ______ by ________
3.)
4.)
Z
S
B
80
5 in
7 in
A
X
Y
A
B
5 in
7 in
80
C
C
R
T
 ______   ______ by ________
 ______   ______ by ________
5.)
6.)
E
E
G
G
F
F
D
D
H
 ______   ______ by ________
H
F is the midpoint of
DG and EH
 ______   ______ by ________
-7-
-8-
7.)
8.)
V
A
T
M
H
U
X
W
 ______   ______ by ________
MT bisects AMH
and ATH
 ______   ______ by ________
9.)
X
Y
Given: WX  YZ , XY  ZW
Prove: WXY  YZW
Z
W
Reasons
Statements
1.)
1.)
2.)
2.)
3.)
3.)
10.)
Given: WX  YZ , WX YZ
X
Y
Prove: WXY  YZW
W
Statements
1.)
1.)
2.)
2.) Reflexive
3.)
4.)
XWY  ZYW
Z
Reasons
3.)
4.)
-8-
-9-
Factoring Review:
1. Collect like terms
2. Factor out any common terms.
Practice:
1.) 5 x 2  5 x  10
2.) x 2  10 x  9
3.) 2 x 2  6 x  14  2 x
4.) 3x 2  5 x  3  x 2  3x  1
5.) 5 x 2  5 x
6.) 3 y 2  24 y  36  18
7.) 2 y 4  10 y 2
8.) 3x 2  18 x  24
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- 10 -
Notes #24: More Proofs and Section 4.4 (Using Congruent Triangles), CPCTC
***_________________ parts of_______________ triangles are _____________ ***
Are the triangles congruent? If so, write the congruence and name the postulate used.
1.)
2.)
X
X
Y
Z
W
Y
Z
W
WX YZ , WX  YZ
WX  YZ , XY  ZW
3.)
4.)
X
W
X
Y
Z
Y
Z
W
WX YZ , XY ZW
WX YZ , XY  ZW
5.) Complete:
B
a) ∆ABC  __________ because _______
b) AB = ____ because ___________
A
E
C
c) AC = EC because ___________. Then C
is the midpoint of _________ by
_______________________________.
D
d) A  _____ because _________. Then AB
ED because _______________________.
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- 11 -
Complete the proofs: follow these key steps
1. Re-draw and label your picture; mark congruencies
2. Find and list 3 congruencies:
shared sides (reflexive)
vertical angles
alternate interior/corresponding angles (only when lines are )
angle bisectors
midpoints
ITT
3. State ∆  ∆ by SSS, SAS, ASA, or AAS
4. State part  part by CPCTC
6.) Given: WX  YZ , XY  ZW
X
Y
Prove: X  Z
Z
W
Statements
Reasons
1.)
1.)
2.)
2.)
3.)  _______   _______
3.)
4.)
4.)
- 11 -
- 12 X
7.) Given: WX YZ , YX WZ
Y
Prove: XY  ZW
Z
W
Reasons
Statements
1.)
1.)
2.)
2.) ________________ angles theorem
3.)
3.)
4.)
4.)  _______   _______
5.)
5.)
8.)
Given: C is the midpoint of AD and BE
D
B
C
Prove: A  D
A
Statements
E
Reasons
1.)
1.)
2.)
2.) Definition of Midpoint
3.)
3.)
4.)  _______   _______
4.)
5.)
5.)
- 12 -
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9.)
Given: CT bisects ACS and ATS
A
Prove: A  S
C
T
Statements
S
Reasons
1.)
1.)
2.)
2.) Definition of __________ ___________
3.)
3.)
4.)
4.)
5.)
5.)
10.)
Given: 1  2, X is the midpoint of WY
Y
Prove: WX  YZ
X
Statements
1
2
Z
W
Reasons
1.)
1.)
2.)
2.) Definition of ______________
3.)
3.)
4.)
4.)
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- 14 -
Factor Review:
1.
2.
3.



Combine like terms
Factor out the greatest common factor if possible
How many terms?
2 terms – Factor using difference of two squares
3 terms – Factor using X and box
4 terms - Factor using grouping
Two terms:
1. Factor our the greatest common factor :
2. If both terms are perfect squares factor into (a  b)(a  b) :
1.) x 2  49
3.) 3x 2  48
5 x 2  125
5( x 2  25)
( x  5)( x  5)
2.) 2 y 2  50
4.)
16 y 2  49
5.) x 2  225
6.) x 4  10  8  x 4
7.) r 4  16
8.) w2  144
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- 15 -
Notes #25: Proof Review:
1.) In equilateral ABC , mA  2 x  4 y and
mB  x  5 y . Solve for x and y.
2.) Solve for x and y
80
x
8
y
50
12
3.) How can you prove triangles congruent?
4.) Solve for x and y
y
3x + 5
120
7x - 3
30
7.) What does CPCTC stand for?
8.) KIM  BEN Complete:
a) IK  _____
b) I  _____
c) ENB  _____
d) IK  _____
- 15 -
- 16 Complete each proof by filling in the blanks.
A
1.
B
Given:
AB || DE
AB  DE
3.
Given:
E is the mdpt
of TP and MR
C
Prove:
∆ABC  ∆EDC
D
2.
2. Alt Int Angles
3.
M
1.
1. Given
2.
TE  PE
2.
____________
theorem
3.
3.
Given:
AB  CD
AB || CD
P
1. Given
4. ∆
3.
∆
4.
5.
2.
R
E
Prove:
TM  PR
E
1.
_____________
T
5.
D
A
4.
Prove:
∆ADB  ∆CBD
B
1.
1.
2.
2. Reflexive
3.
3. Alt Int. Angles
C
Given:
1  4; 2  3
M is the mdpt.
of AB
Prove:
AC  BD
A
C
1
D
2
3
M
1.
1.
2. AM  BM
3.
3. ∆
3.
4
B
theorem
4.
4.
4.
∆
4.
- 16 -
17
5.
Given:
AD || ME
MD || BE
M is the mdpt.
of AB
Prove:
MD  BE
A
1
D
2
M
3
1.
2.
3.
3.
∆
S
1
2
3
T
4
E
B
2. 2  4
R
Given:
RS  RT
Prove:
3  4
4
1.
4. ∆
7.
1.
1. Given
2.
3. ITT
3. 3  1
4  2
2.
4.
4.
4.
5.
5.
G
6.
Given:
WO  ZO
XO  YO
Prove:
W  Z
W
Z
X
1.
1.
2.
2.
3. ∆
4.
∆
3.
4.
O
8.
Prove:
JG  MK
Y
1
Given:
M is the mdpt
of JK
1  2
2
J
M
K
1.
1.
2. KM  JM
2.
3. JM  JG
3.
4.
4.
17
18
Algebra Review
Factoring Review:
Three terms;
1. divide all terms by common denominator
2. Put quadratic in standard form; Coefficient of x2 must be 1.
3. find factors of last term that will add up to middle coefficient
Practice:
1.) x 2  5 x  6
2.) x 2  10 x  9
3.) x 2  5 x  14
4.) x 2  2 x  8
5.) 5 x 2  5 x  10
6.) 3 y 2  24 y  36
7.) 2 y 2  10 y  8
8.) 3x 2  18 x  24
2x2 – 16x + 30
2(x2 – 8x +15)
2(x2 – 8x +15)
2(x – 5)(x – 3)
18
19
Notes 27: Section 4.6 (Congruence in Right Triangles) Section 4.7( Using Corresponding
Parts of Congruent Triangles)
HL:
___________ - __________ -(
)Postulate
Hypotenuse: Side opposite the right angle
B
Leg: Side adjacent to right angle
C
A
Which of these triangles are congruent?
Using the HL Postulate:
1.)  _______   _______ by _______
A
D
B
C
19
20
V
2.)  _______   _______ by _______
U
X
W
Given: CA  ED, AD is the perpendicular bisector of CE .
Prove: CBA  EBD
D
C
B
A
Statements
1.
2.
3.
4.
5.
CBA and EBD
CBA  EBD
are rt.
E
Reasons
1. Given
2.
3. Def. of  bisector; Def. of midpoint
4. Def. of right triangles
5.
20
21
Proving Overlapping Triangles Congruent
For #1-5, complete the following:
a) Separate the overlapping triangles. Mark the side or angle that is/was overlapping.
b) Mark the congruent segments and congruent angles.
c) Are the triangles congruent? If yes, state the postulate used to state the triangle congruence
(SSS, SAS, ASA, AAS, or HL)
1.)
W
Z
Y
X
WX  ZY , WXY  ZYX
2.)
A
B
D
C
AC  DB
3.)
M
N
L
O
L  O, LMN  ONM
21
22
4.)
A
E
B
D
C
AC  EC , A  E
5.)
B
A
E
D
C
(the triangles to examine are
ABD and ABC )
AC  BD
For #6-9, complete the following proofs:
6.) Given:
Prove:
WX  ZY , WXY  ZYX
W
Z
WXY  ZYX
Y
X
Statements
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
22
23
7.) Given: AC  DB
Prove:
A
B
D
C
ABC  DCB
Statements
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
8.)
Given: L  O, LMN  ONM
M
N
Prove: LM  ON
L
Statements
O
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
23
24
9.)
Given: AC  EC , A  E
A
E
Prove: AD  EB
B
D
C
Reasons
Statements
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
24
25
Complete each proof by filling in the blanks.
1.
O
Given:
B
DB  DN
OD bisects BDN
1 2
3 4
S
Prove:
3  4
5 6
D
1. DB  DN
1. Given
2. OD bisects BDN
3. Given
3.
2.
4.
4.
5. OBD  OND
5.
6.
6.
7.
7.
8.
8.
9. OBS  ONS
9.
10.
10.
N
2.Given:
NGI  NAI
1  2
Prove:
GT  AT
1. NGI  NAI
G
N
A
3
T
4
1
I
2
1. Given
2.
2.
3. 1  2
3. Given
4. GIN  AIN
4.
5.
5.
6.
6.
7.
7.
8. GTN  ATN
8.
9.
9.
25
26
3.
Given:
BU  CH
UC  HB
1  2
Prove:
UE  HL
U
C
4.
L
1
E
B
2
H
Given:
UC || HB
UB || HC
BE  CL
Prove:
1  2
U
C
L
1
2
E
B
H
1. BU  CH
1. Given
1. UC || HB
1. Given
2. UC  HB
2. Given
2.
2.
3.
3.
3.
3.
4. BUC  CHB
4.
4. UB || HC
4. Given
5. BCU  CBH
5.
5.
5.
6. 1  2
6. Given
6. BUC  CHB
6.
7. UCE  HBL
7.
7.
7.
8.
8.
8. BE  CL
8. Given
9. BUE  CHL
9.
10.
10.
26
27
2
Algebra Review: Factoring quadratics with an x coefficient not equal to 1.
1. Put in standard form
2. Divide by greatest common factor if possible
3. Use X and box to factor
2 x 2  13x  20
Practice:
1.) 2 x 2  7 x  3
2.) 4 x 2  12 x  5
3.) 3x 2  7 x  6
4.) 15 x 2  7 x  2
5.) 6 x 2  7 x  3
6.) 2 x 2  5 x  3
27
28
Notes 28: Chapter 4 Review:
Proof Review
C
1.) Given: X is the mdpt. of CB
CD AB
D
X
A
B
Prove: AB  DC
1.
2.
_____
____
1.
___________
2.
_____
3.
3.
4.
4.
5.
5.
2.
1.
Given
L
M
Given: LM  JK
LM || JK
Prove: JM  LK
J
LM  JK; LM || JK
1.
K
Given
2.
2. Alt. int. angle thm.
3.
3.
4.
_____________
5.
4.
5.
3.) Given: WX  ZY , WY  ZX
W
X
3
4
Prove: WX ZY
1
Y
2
Z
1.__________________________
1._________________________
2.__________________________
2._________________________
3.__________________________
3._________________________
4.__________________________
4._________________________
5.__________________________
5._________________________
28
29
O
4.) Given: 1  3
Prove: ON  OP
N
1
2
P
3
1.__________________________
1.__________________________
2.__________________________
2.__________________________
3.__________________________
3.__________________________
4.__________________________
4.__________________________
Are the triangles congruent? If so, write the congruence and name the postulate used.
5.)
6.)
E
E
G
G
F
F
D
D
H
H
 ______   ______ by ________
7.)
F is the midpoint of
DG and EH
 ______   ______ by ________
8.)
V
A
T
M
H
U
MT bisects AMH
and ATH
W
X
 ______   ______ by ________
9.)
 ______   ______ by ________
10.)
V
A
B
D
C
 ______   ______ by ________
U
X
W
 ______   ______ by ________
29
30
Algebra Review
1.) Factor: 3 x 2  6 x  12
2.) Factor: 5 x 2  125
3.) Factor: x 2  11x  18
4.) Factor: x 2  1x  12
5.) Factor: 6 x 2  7 x  3
6.) Factor: 3 x 2  4 x  4
7.) Factor: 3 x 2  5 x  2
8.) Factor:
3 x 2  15 x
30
31
9.) Factor: y  49
2
10.) Factor: x  4 x  32
2
31
32
Chapter 4 Study Guide:
1. Given: WX ZY , XY WZ
X
Y
3
4
Prove: X  Z
2
1
W
Z
Statements
1. _________________________
2. _________________________
_________________________
3. _________________________
4. _________________________
5. _________________________
Reasons
1. ________________________
2. ________________________
________________________
3. ________________________
4. ________________________
5. ________________________
X
Y
3
2. Given: WX  ZY , XY  WZ
4
2
Prove: XY WZ
W
1
Z
Statements
1. _________________________
2. _________________________
3. _________________________
4. _________________________
5. _________________________
Reasons
1. ________________________
2. ________________________
3. ________________________
4. ______ CPCTC__________
5. ________________________
B
D
3. Given: AB DE , C is the midpoint of BE
C
Prove: AC  CD
A
Statements
1. _________________________
2. _________________________
3. _________________________
4. _________________________
5. _________________________
6. _________________________
E
Reasons
1. ________________________
2. ________________________
3. ________________________
4. ________________________
5. ________________________
6. ________________________
N
P
4. Given: NO  PO, MO  QO
O
Prove: M  Q
Statements
1. _________________________
2. _________________________
3. _________________________
4. _________________________
M
Q
Reasons
1. ________________________
2. ________________________
3. ________________________
4. ________________________
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ABC is equilateral. If mA  2 x  y and mB  4 x  y , solve for x and y.
5.
6. In
XYZ , XY  YZ . If mX  5x 10 and mZ  2x  44 solve for mX
7. Are the pairs of triangles congruent? If so, name the congruence and the postulate used.
a)
b)
c)
d)
8. a) Solve for x:
b) Solve for y:
B
B
3y - 6
30
2x + 17
2y + 8
A
12
64
58
6x - 7
C
C
A
9. Factor:(Show work on separate sheet of paper)
a) x 4  81
b) y 2  16 y  64
c) x 2  5 x  36
d) 3x 2  27
e) 4 x 2  25
f) 12 x 2  4 x  40
g) 3 x 2  7 x  4
h) 36 x 2  48 x  15
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