Download 4.1 Worksheet

-1-
Geometry Rules!
Chapter 4 Notes
Notes #22: Section 4.1 (Congruent Triangles) and Section 4.4 (Isosceles Triangles)
Congruent Figures
Corresponding Sides
Corresponding Angles
***_________________ parts of_______________ 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:
-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 _________.
-3-
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.
10.) What can you conclude from the picture?
9.) In equiangular ∆ABC, AB = 2x + y,
BC = 6x – 2y, and AC = 10. Solve for
x and y.
-4F
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
4
Reasons
Statements
1.)
1.)
2.)
2.)
C
-5-
3.)
3.) Substitution
4.)
4.)
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Notes #23: Sections 4.2 and 4.5 (Methods of Proving Triangles Congruent)
Q: How can we prove that two triangles are congruent to each other?
A: Five ways: SSS, SAS, ASA, AAS, HL
SSS:
_______-________-________ Postulate
SAS:
_______-________-________ Postulate
ASA:
_______-________-________ Postulate
AAS:
_______-________-________ Postulate
HL:
_____________-______________-(_______________) Postulate
-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, AAS, or HL
 Look for “hidden” pieces in:
- vertical angles
- overlapping sides
- congruent angles formed by parallel lines
- bisected angles
- ITT/Converse of ITT
- midpoints
2.)
1.)
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 ________
-8-
7.)
8.)
V
A
T
M
H
U
MT bisects AMH
and ATH
W
X
 ______   ______ by ________
9.)
 ______   ______ by ________
10.)
V
A
B
D
U
C
 ______   ______ by ________
W
X
 ______   ______ by ________
11.)
X
Y
Given: WX  YZ , XY  ZW
Prove: WXY  YZW
Z
W
Reasons
Statements
1.)
1.)
2.)
2.)
3.)
3.)
-9-
12.)
X
Y
Given: WX  YZ , WX YZ
Prove: WXY  YZW
Z
W
Reasons
Statements
1.)
1.)
2.)
2.) Reflexive
3.)
XWY  ZYW
3.)
4.)
4.)
Notes #24: More Proofs and Section 4.3 (Using Congruent Triangles)
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
W
WX YZ , XY ZW
WX YZ , XY  ZW
Y
Z
- 10 -
5.) Complete:
B
a) ∆ABC  __________ because _______
b) AB = ____ because ___________
E
C
A
c) AC = EC because ___________. Then C
is the midpoint of _________ by
_______________________________.
D
d) A  _____ because _________. Then AB
ED because _______________________.
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, AAS, or HL
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.)
- 11 -
4.)
7.) Given: WX YZ , YX WZ
4.)
X
Y
Prove: XY  ZW
Z
W
Reasons
Statements
1.)
1.)
2.)
2.) If two parallel lines are cut by a
transversal, then ____________________
angles are congruent.
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.)
- 12 -
5.)
5.)
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.)
- 13 -
Notes #25: Algebra and Proof Review:
y
1.) Graph the points and name the quadrant
in which each point is found:
10
9
8
7
6
A(-3, 2)
5
B(0, -7)
4
3
2
C(4, -1)
D(6, 0)
1
x
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
2.) Evaluate for a = -2 and b = 3
-4
-5
-6
-a – b(ab – 3)
-7
2
-8
-9
-10
4.) In equilateral ABC , mA  2 x  4 y and
mB  x  5 y . Solve for x and y.
3.) Simplify:
-32 – 4(22 – 1) – (-3)
5.) Solve for x and y
y
80
x
6.) Solve for x and y
3x + 5
8
120
y
50
12
7x - 3
30
7.) What does CPCTC stand for?
8.) KIM  BEN Complete:
a) IK  _____
b) I  _____
c) ENB  _____
d) IK  _____
- 14 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
T
M
1.
2.
2. ll lines cut by a
_____________
2.
4. ∆
A
D
B
1.
1.
2.
2. Reflexive
3.
3. ll lines cut by a
trans form  alt. int.
’s
4.
4.
3.
∆
4.
5.
4.
Prove:
∆ADB  ∆CBD
2.
3.
trans form  alt. int.
3.
Given:
AB  CD
AB || CD
TE  PE
____________
’s
3.
1. Given
2.
1. Given
P
Prove:
TM  PR
E
1.
R
E
C
5.
Given:
1  4; 2  3
M is the mdpt.
of AB
Prove:
AC  BD
A
C
1
2
1.
2. AM  BM
3.
3. ∆
3.
4.
3
M
1.
∆
D
4.
4
B
15
5.
Given:
AD || ME
MD || BE
M is the mdpt.
of AB
Prove:
MD  BE
2
M
3
2.
S
1
2
3
T
4
1.
1. Given
2.
3. ITT
3. 3  1
4  2
2.
4.
4.
3.
∆
4.
5.
5.
Given:
WO  ZO
XO  YO
Prove:
W  Z
G
W
Z
X
1.
1.
2.
2.
4.
E
R
Given:
RS  RT
Prove:
3  4
1.
3.
3. ∆
7.
D
B
2. 2  4
6.
1
4
1.
4. ∆
A
∆
3.
4.
O
8.
Y
1
Given:
M is the mdpt
of JK
1  2
Prove:
JG  MK
2
M
K
1.
1.
2. KM  JM
2.
3. JM  JG
3.
4.
4.
J
16
Notes #26: Section 4.7 (Special Segments in Triangles) and Proof Review
Median: connects a ___________ to the __________ of the opposite side
________ and _________ are medians of
__________
B
C
A
_____is the _____________ of AC
_____ is the _____________ of BC
______ = ______
B
______  ______
C
A
____ is equidistant from ___ and ___
____ is equidistant from ___ and ___
Altitude: a ________________ segment from a vertex to an opposite side
________ and _______ are ______________
of ∆ABC
B
C
A
B
A
C
____

_____
____

_____
m _________  90
m _________  90
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Perpendicular Bisector: a _______________________ segment to the
________________of the opposite side
______ and ______ are _________________
_________________ of ∆ABC
B
_____is the _____________ of AC
C
A
_____is the _____________ of BC
____  _____
____ = _____
B
C
A
____

_____
____

_____
If X is on the perpendicular bisector to AC
then X is equidistant from ______
and
_______.
Angle Bisector: cuts a ___________ ____________ into two equal
____________ AND is ________________________ from the sides of the
angles.
__________ are _____________________ of
∆ABC
B
m _______  m _________
C
A
 _______   _________
_____ is equidistant from _____ and ______.
B
_____ is equidistant from ______ and ______.
A
C
18
Practice:
1.) XZ is a median to WY . WZ  5x  3 and
ZY  22 . Solve for x.
X
W
Y
Z
2.) BD is a perpendicular bisector to AC .
mBDA  3x 15 , AD  2 y  6 and
DC  4 y  14 . Solve for x and y.
B
A
C
D
3.) Name an angle bisector, a median, and an
altitude of ABC
C
Angle bisector: _________
X
Median: _________
Y
Altitude: _________
A
Z
B
19
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
Prove: JM  LK
J
LM  JK; LM || JK
1.
2.
L
M
Given: LM  JK
LM || JK
K
Given
2. If parallel lines are cut by a transversal,
then alternate interior angles are congruent.
3.
4.
3.
_____________
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._________________________
20
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 ________
21
Notes #27: Test and Algebra Review
1.) Solve:
2.) Get y alone:
3
 x  1  x  3
5
3.) Add:
2x  5 y  8
4.) Subtract:
6x
2
 4 x  3   2 x 2  3x  1
5.) Simplify:
6x
7.) Evaluate: if a = -2 and b = 4
-2xy(4x2 + 6y - 3)
8.) Evaluate: if x = -1 and y = 3
-2a2 – 3ab(a + 2b)
11.) FOIL:
 4 x  3   2 x 2  3x  1
6.) Distribute:
2 x(4 x 2  x)  5x 2 ( x  7 x)
9.) FOIL:
2
( x  3)( x  4)
(5 x  2)(2 x  1)
-y2 – xy(y – x)
10.) FOIL:
12.) FOIL:
(3x  1)(2 x  4)
3 x( x  4)( x  1)
22
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. ________________________
23
5. Name each of the special segments:
(a) median
C
B
(b) altitude
D
(c) angle bisector
E
A
6. Complete:
(a) F is equidistant from _____ and _____.
Therefore, _____  ________
F
(b) Any point on AD is equidistant from
______ and _______.
ABC is equilateral. If mA  2 x  y and mB  4 x  y , solve for x and y.
7.
8. In
XYZ , XY  YZ . If mX  5x 10 and mZ  2x  44 solve for mX
9. Are the pairs of triangles congruent? If so, name the congruence and the postulate used.
a)
b)
c)
d)
10. 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
24
11.) Solve each system…
a) Using Elimination
x + 2y = 2
3x – 3y = -12
b) Using Substitution
12.) The sum of twice an angle’s complement
and it supplement is 249 degrees. Find the angle,
its complement, and its supplement.
13.) The difference of twice a number and eight
is four more than three times the number. Find
the number.
14.) QX bisects PQR , mPQX  4 x  11 ,
mXQR  2 x  5 . What kind of angle is PQR ?
15.) The measure of one angle of a triangle is ten
more than twice the smaller angle. The third angle
of the triangle is ten less than six times the
smallest angle. Find the measure of all three
angles and then classify the triangle. (Hint: let x =
the smallest angle)
16.) Evaluate
a) (p – px) + (a + p) for a = -3, p = 2, x = -4
x – 3y = -4
2x + 2y = 4
b) -32 – 2[2(5 – 3) – (2 – 3)]
17.) If AH is the perpendicular bisector of MT ,
solve for x and y.
mMHA  4x  18
MH  5 y  3
HT  4 y  15
A
M
H
T
25