Theorems 4 – 18 &
more definitions, too!
Page 104, Chapter Summary: Concepts and Procedures
After studying this CHAPTER, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
2.2 Recognize complementary and supplementary angles
2.3 Follow a five-step procedure to draw logical conclusions
2.4 Prove angles congruent by means of four new theorems
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
2.6 Apply the multiplication and division properties of segments and angles
2.7 Apply the transitive properties of angles and segments
2.7 Apply the Substitution Property
2.8 Recognize opposite rays
2.8 Recognize vertical angles
2
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
Related Vocabulary
COORDINATES
OBLIQUE
LINES
ORIGIN
X-axis
PERPENDICULAR
Y-axis
3
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
Related Vocabulary
PERPENDICULAR – lines,
rays, or segments that
INTERSECT at right angles
OBLIQUE LINES – when
DEFINITIONS
lines, rays, or segments
INTERSECT and are
NOT PERPENDICULAR
4
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
CHAIN REASONING
Related Vocabulary
SYMBOLS:
Given:
OH  OK
If
OH  OK ,
RIGHT ANGLE
then
∡HOK is a Rt ∡
NOT PERPENDICULAR
and if
∡HOK is a Rt ∡,
then
m∡HOK = 90

H
⊬
PERPENDICULAR
O
K
CONDITIONAL
If a right angle is created at the intersection of two rays, then the rays are perpendicular!
CONVERSE
If two rays are perpendicular, then they create a right angle!
5
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
CHAIN REASONING
Related Vocabulary


Right
Angles
H
90⁰
90⁰
O
Given:
m∡ HOK = 90
If
m∡HOK = 90
then
∡HOK is a Rt ∡
and if
∡HOK is a Rt ∡,
then
OH  OK
K
CONDITIONAL
If a right angle is created at the intersection of two rays, then the rays are perpendicular!
CONVERSE
If two rays are perpendicular, then they create a right angle!
6
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
90⁰
Right ∡
Perpendicularity,
right angles, and
H
90⁰ measurements
all go together!

Right ∡
90⁰

90⁰
O
K
CONDITIONAL
If a right angle is created at the intersection of two rays, then the rays are perpendicular!
CONVERSE
If two rays are perpendicular, then they create a right angle!
7
Chapter 2, Section 1: “Perpendicularity”
After studying this SECTION, you should be able to . . .
2.1 Recognize the need for clarity and concision in proofs
2.1 Understand the concept of perpendicularity
Related Vocabulary
y-axis
ORIGIN
5
4
HH (0, 3)
COORDINATES
3
2
G (-4,
G 0)
x-axis
Can you name
the  lines?
-5
-4
-3
1
-2
C
(-3,
-2)
2)
(3,-2)
2)
DBA(-3,
(3,
-1
0 1
E (0,-1 0)
2
3
4
Could
any lines drawn be
“oblique lines”?
F
F (4, 0)
5
-2
-3
J J-4(0, -3)
Can you name
the ‖ lines?
‖  parallel
-5
Remember: The x-axis is  to the y-axis
8
2.1 Example
Find the area of rectangle PACE
Remember an important property
of rectangles is that
pairs of
AreaBOTH
RECT = (length)(width)
opposite sides are congruent, and:
Given: AP ‖ to the y-axis
CE ‖ to the y-axis
AreaRECT = (7 units)(width)
(4 units)
If two segments are congruent,
Areathe=SAME
28 units2
then they have
7
measure!
RECT
Width = |y – y|
5
Length = |x – x|
Width = |2 – (-2)|
4
Length = |3 – (-4)|
3
A
Width = |2 + 2|
Width = |4|
4
-5
-4
P
-3
-2
-1
C
2
Length = |3 + 4|
1
Length = |7|
0 1
-1
-2
-3
2
3
4
5
E
-4
-5
9
Chapter 2, Section 2: “Complementary and Supplementary Angles”
After studying this SECTION, you should be able to . . .
2.2 Recognize complementary and supplementary angles
Related Vocabulary
COMPLEMENT
(NOT the same as: “You look very nice today!”)
COMPLEMENTARY ANGLES
SUPPLEMENT
(NOT THE SAME AS: “Did you take your vitamins today!”)
SUPPLEMENTARY ANGLES
10
Chapter 2, Section 2: “Complementary and Supplementary Angles”
QUESTION!
After studying this SECTION, you should be able to . . .
2.2 Recognize complementary and supplementary angles
If two anglesRelated
areVocabulary
COMPLEMENTARY ANGLES,
COMPLEMENT
(then) are they also ADJACENT ANGLES?
- the NAME given to each of the two angles whose sum equals 90⁰
COMPLEMENTARY ANGLES - two angles whose sum equals a 90⁰ right angle
V
15⁰
N
V
30⁰
V
60⁰
75⁰
A
N
A
N
11
Chapter 2, Section 2: “Complementary and Supplementary Angles”
QUESTION!
After studying this SECTION, you should be able to . . .
2.2 Recognize complementary and supplementary angles
Related Vocabulary
If two angles are SUPPLEMENTARY ANGLES,
SUPPLEMENT
(then) are they also ADJACENT ANGLES?
- the NAME given to each of the two angles whose sum equals 180⁰
SUPPLEMENTARY ANGLES - two angles whose sum equals a 180⁰ straight angle
85⁰
R
A
130⁰
R
112⁰18’40”
67⁰41’20”
50⁰
95⁰
T
R
A
T
P
T
12
Chapter 2, Section 2: “Complementary and Supplementary Angles”
After studying this SECTION, you should be able to . . .
THINK –
Is the answer
reasonable?
Related Vocabulary
If two angles are
Is one of the angles 15
complementary
angles,
The measure of one of two complementary
angles is 15 more
than twice the other.
more
than
twice
the
Find the measure of each angle.
then their sum equals
other?
90!
2.2 Recognize complementary and supplementary angles
x + 2x + 15 = 90  Write equation
3x + 15 = 90  Simplify
3x = 75  Solve for x
50
2x75⁰
++ 15
15
25⁰
x
x = 25  Substitute
YES!
13
If a problem contains ONLY complements or ONLY supplements,
use the previous method.
Begin by drawing a right angle for two complementary angles
or a straight angle to model two supplementary angles,
and label them according to the information given in the problem!
HOWEVER,
if a problem refers to BOTH the complement AND the supplement
in the same problem ,
use the NEXT method:
14
Chapter 2, Section 2: “Complementary and Supplementary Angles”
After studying this SECTION, you should be able to . . .
2.2 Recognize complementary and supplementary angles
Use the “Boxer” Method to write expressions for each type of angle:
Are you wondering, “what is the “Boxer Method”?”
Well, first make a “BOX,” and then let “the angle” equal x
THE ANGLE
x⁰
30⁰
Complements
60⁰
COMPLEMENT
(90 – x)⁰
60⁰
30⁰
x⁰
Supplements
SUPPLEMENT
(180 – x)⁰
150⁰
150⁰
30⁰
x⁰
15
Chapter 2, Section 2: “Complementary and Supplementary Angles”
2.2 Recognize complementary and supplementary angles
Example
The measure of the supplement of an angle is 60 less than 3 times the
complement of the angle.
√
Find the measure of the complement.
The measure of the supplement of an angle is 60 less than 3 times the complement
(180 – x)
ANGLE
COMP
SUPP
x
3(90 – x) - 60
=
15⁰
90 – x
75⁰
180 – x
165⁰
180 – x = 270 -3x -60
x “the angle”
15
180 + 2x = 210
2x = 30
90 – 15
x
x
15
180 –– 15
x
180
Complement
x
15
x = 15
Supplement
16
Chapter 2, Section 3: “Drawing Conclusions”
After studying this SECTION, you should be able to . . .
2.3 Follow a five-step procedure to draw logical conclusions
Related Vocabulary
No NEW vocabulary!
17
Chapter 2, Section 3: “Drawing Conclusions”
After studying this SECTION, you should be able to . . .
2.3 Follow a five-step procedure to draw logical conclusions
See very important TABLE on page 72!
5-STEP Procedure for Drawing Conclusions:
•1. MEMORIZE theorems, definitions, and postulates
• 2. Look for KEY WORDS and SYMBOLS in the “givens”
• 3. Think of all the theorems, definitions, and postulates that involve those keys.
• 4. Decide which theorem, definition, or postulate allows you to draw a conclusion
• 5. DRAW A CONCLUSION, and give a reason to justify it.
NOTE: The “If . . .” part of the reason should match the GIVEN information!
AND
the “then . . .” part matches the CONCLUSION being justified!
CAUTION! Be sure not to reverse that order!!!
18
Chapter 2, Section 3: “Drawing Conclusions”
After studying this SECTION, you should be able to . . .
PRACTICE EXAMPLES
then____?______
. . . AB ≅ BC
1) If B bisects AC, then
A
B
Key info: a point, bisect, and seg
C
B
then _____?_______.
∡BAC is a Rt ∡
2) If AB  AC, then
A
Key info: ,, and 
C
A
C
3) If ∡ABC ≅ ∡CBD ≅ ∡DBE,
D
info:
≅ ∡ ∡ABE
then .Key
____?____.
then
. . BC
and∡ ≅
BD∡trisect
B
E
19
Chapter 2, Section 3: “Drawing Conclusions”
After studying this SECTION, you should be able to . . .
JUSTIFY your CONCLUSIONS!
then____?______
. . . AB ≅ BC
1) If B bisects AC, then
A
B
REASON: If a seg is bisected by a point, then
the seg is divided into two congruent segs
C
B
then _____?_______.
∡BAC is a Rt ∡
2) If AB  AC, then
A
REASON: If two rays are perpendicular,
then they form a right angle
A
C
3) If ∡ABC ≅ ∡CBD ≅ ∡DBE,
then
then
. . ____?____.
. BC and BD trisect ∡ABE
REASON: If an angle has been divided
into 3 congruent angles,
then it was trisected by two rays.
C
D
B
E
20
Chapter 2, Section 4: “Congruent Supplements and Complements”
After studying this SECTION, you should be able to . . .
2.4 Prove angles congruent by means of four new theorems
Related Vocabulary
No NEW vocabulary!
BUT . . .
THEOREM #7
THEOREM #6
THEOREM #5
THEOREM #4
21
Chapter 2, Section 4: “Congruent Supplements and Complements”
After studying this SECTION, you should be able to . . .
2.4 Prove angles congruent by means of four new theorems
THEOREM #4
If angles are supplementary to the same angle,
then they are congruent
∡1 is supplementary to ∡G
∡2 is also supplementary to ∡G
What can we conclude about ∡1 and ∡2?
=
160⁰
120⁰
60⁰ 2
G
22
Chapter 2, Section 4: “Congruent Supplements and Complements”
After studying this SECTION, you should be able to . . .
2.4 Prove angles congruent by means of four new theorems
THEOREM #5
If angles are supplementary to congruent angles,
then they are congruent
∡G is supplementary to ∡E
∡O is supplementary to ∡M
∡E ≅ ∡O
M
130⁰
130⁰
50⁰
G E
50⁰
O
What can we conclude about ∡G and ∡M?
23
Chapter 2, Section 4: “Congruent Supplements and Complements”
After studying this SECTION, you should be able to . . .
2.4 Prove angles congruent by means of four new theorems
THEOREM #6
If angles are complementary to the same angle,
then
they
are
What
can
wecongruent
conclude?
THEOREM #7
If angles are complementary to congruent angles,
then
they
What
canare
we congruent
conclude?
The only difference is the sum! (90 versus 180)
24
Chapter 2, Section 4: “Congruent Supplements and Complements”
After studying this SECTION, you should be able to . . .
S
Complete a Proof!
Given:
∡1 is comp to ∡4
∡2 is comp to ∡3
?
3
R
4
?
RT bisects ∡SRV
PROVE:
TR bisects ∡STV
1
2
T
V
Statements
Reasons
1) ∡1 is comp to ∡4
1) Given
2) ∡2 is comp to ∡3
2) Given
3) RT bisects ∡SRV
3) Given
4) ∡3 ≅ ∡4
4) If a ray bis an ∡, it div it into 2 ≅ ∡s
5) ∡1 ≅ ∡2
5) If ∡’s comp ≅ ∡s, then they are ≅
6) TR bisects ∡STV
6) If an ∡ is div into 2 ≅ ∡s, then
it was bisected by a ray!
25
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
If asubtraction
segment
is added
to twoand
congruent
2.5 Apply the
properties
of segments
angles
segments,
the
sums
are
congruent.
Related Vocabulary
(Addition Property)

A
Note that we first need to know that two segments are
7cm
congruent, Band then
7 cm
3cm that
C we are adding
D
 the SAME segment to both of them.
AC = BD, because
(Commutative Property
of Addition!)
(7) + (3) = (3) + (7)
AB + BC = BC + CD, 
If two segments have the same measure,
they are congruent!
AC  BD
26
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
If the
an subtraction
angle is added
congruent
2.5 Apply
propertiesto
of two
segments
and anglesangles,
then theRelated
sumsVocabulary
are congruent.
(Addition Property)A mABC = 50.03
(Commutative Property of Addition!)

C
Note that we first need to know that two angles
are
and+then
50 + ∡CBD congruent,
=
∡CBD
50 that we are adding
m∡ABC + m∡CBD
m∡CBDangle
+ m∡DBE
 =
the SAME
to both of them.
D
m∡ABD =
m∡CBE, so
B
ABD  CBE
If two angles have the same measure,
they are congruent!
mDBE = 50.03
E
27
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5
If an angle is subtracted from two congruent
Apply the subtraction properties of segments and angles
angles, the differences are congruent.
Related Vocabulary
(Subtraction
Property) m∡ABD = 80⁰

80
A mABC = 50.03
Note that we first need to know that two angles
are
C
congruent,
that we are subtracting
- ∡CBD
- ∡CBD
= 80and then

m∡ABD - m∡CBD
the
angle
from both of them.
= SAME
m∡CBE
- m∡CBD
m∡ABC =
m∡DBE, so
B
 ABC   DBE
If two angles have the same measure,
they are congruent!
D
mDBE
m∡CBE==50.03
80⁰
E
28
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
F
If congruent segments areCadded to congruent
the sums are congruent.
CF + FG = segments,
DE + EH
(Addition Property)
D
E
G
H
CG
DH,
Note=that
firstso
we need 2 congruent segments,
then we need 2 different congruent segments
CG ≅ DH to ADD.
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
J
m∡JIL
+ m∡LIKangles
= m∡LKI
+ m∡JKL
If congruent
are added
to congruent
angles, the sums are congruent.
(Addition Property)
L
Note that first we need 2 congruent
angles, then we need to add two different
I
JIK congruent
JKI angles
30
K
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
10
If
a
segment
(or
angle)
is
subtracted
from
QR - BR = BA - BR
Q
B
R
congruent segments (or angles), the
10
differences are congruent.
(Subtraction Property)
QB ≅ RA
A
Note that we need to start with congruent
angles or segments and then subtract the
same angle or segment from both.
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
If a segment (or angle) is subtracted from congruent
segments (or angles), the differences are congruent.
(Subtraction Property)
Note that we need to start with congruent angles or segments
and then subtract the same angle or segment from both.
A
mABD = 78
C
mABD - mCBD = mCBE - mCBD
D
mCBE = 78
B
E
ABC DBE
Chapter 2, Section 5: “Addition and Subtraction Properties”
After studying this SECTION, you should be able to . . .
2.5 Apply the addition properties of segments and angles
2.5 Apply the subtraction properties of segments and angles
If congruent segments (or angles) are subtracted
from congruent segments (or angles), the
differences are congruent. (Subtraction Property)
Note that we start with congruent segments or angles, and
then subtract congruent segments or angles.
S
U
W
T
V
mSTV = mUVT = 130
mWTV =mWVT = 30
mSTV - mWTV = mUVT - mWVT
∡STW ≅ ∡UVW


An addition property is used when the
segments or angles in the conclusion are
greater than those in the given information
A subtraction property is used when the
segments or angles in the conclusion are
smaller than those in the given information.
Theorem: If a segment is added to two congruent
segments, the sums are congruent. (Addition Property)
Given:
PQ  RS
P
Q
R
S
Conclusion: PR  QS
Statements
Reasons
1.
1. Given
PQ  RS
2. PQ = RS
2. If two segments are congruent,
then they have the same measure
3. PQ + QR = RS + QR
3. Additive Property of Equality
4. PR = QS
4. Addition of Segments
5.
5. If two segments have the same
measure then they are congruent
PR  QS
How to use this theorem in a proof:
Given:
GJ  HK
Conclusion:
GH  JK
Statements
1.
2.
M
G
?
H
J
?
K
Reasons
GJ  HK
1. Given
GH  JK
2. If a segment is subtracted
from congruent segments, then
the resulting segments are
congruent. (Subtraction)

If segments (or angles) are congruent, then
their like multiples are congruent.








A
B
C
D
E
F
G
H
Example: If B, C, F, and G are trisection points and AB  EF ,
then AD  EH by the Multiplication Property.
If segments (or angles) are congruent, then their like divisions are
congruent.
D
C
S
A
T
Z
O
G
If ∡CAT ≅ ∡DOG, and AS and OZ are angle bisectors
then, ∡CAS ≅ ∡DOZ by the division property

Look for the DOUBLE USE of the words
midpoint, trisects, or bisects in the “Givens.”

Use MULTIPLICATION if what is Given is less
than the Conclusion

Use DIVISION if what is Given is greater than
the Conclusion
Given: MP  NS
O is the midpoint of MP
R is the midpoint of NS
Prove: MO  NR
Statements
1.
2.
3.
4.
5.
5.
MP ≅ NS
O is mdpt of MP
MO ≅ OP
R is mdpt of NS
NR ≅ RS
MO ≅ NR

M


O
P


N
R

S
Reasons
1.
2.
3.
4.
4.
5.
Given
Given
A mdpt divides a seg into 2 ≅ segs
Given
Same as #3
If segs are ≅, then their like divisions are ≅
(DIVISION PROPERTY)
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
Related Vocabulary
SUBSTITUTE
SUBSTITUTION
Theorems
Theorem 16
Theorem 17
41
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
A
AB ≅ BC
B
BC ≅ CD
CONCLUSION?
AB ≅ CD
C
D
THEOREM:
If segments are congruent to the SAME segment,
then they are congruent to each other.
42
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
∡1 ≅ ∡2
2
∡2 ≅ ∡3
CONCLUSION?
3
1
∡1 ≅ ∡3
THEOREM:
If angles are congruent to the SAME angle,
then they are congruent to each other.
43
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
AB ≅ NM
A
QR ≅ MP
NM ≅ MP
CONCLUSION?
AB ≅ CD
N
B
M
R
Q
P
THEOREM:
If segments are congruent to congruent segments,
then they are congruent to each other.
44
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
∡7 ≅ ∡5
∡6 ≅ ∡8
∡5 ≅ ∡6
CONCLUSION?
7
∡7 ≅ ∡8
5
6
8
THEOREM:
If angles are congruent to congruent angles,
then they are congruent to each other.
45
Chapter 2, Section 7: “Transitive and Substitution Properties”
After studying this SECTION, you should be able to . . .
2.7 Apply the Transitive Property of angles and segments
2.7 Apply the Substitution Property
Given:
∡1 comps ∡2
1
∡2 ≅ ∡3
2
3
m∡1 + m∡2 = 90
m∡2 ≅ m∡3
∴ m∡1 + m∡3 = 90
By Substitution Property!
46
Chapter 2, Section 8: “Vertical Angles”
After studying this SECTION, you should be able to . . .
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
Opposite Rays -
(definition) – collinear rays that share a common endpoint
and extend in opposite directions
Vertical Angles- (definition) – two angles whose sides are formed by opposite rays.
THEOREM 18
Vertical angles are CONGURENT!
47
Chapter 2, Section 8: “Vertical Angles”
After studying this SECTION, you should be able to . . .
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
Opposite Rays - (definition) – collinear rays that share a common endpoint
and extend in opposite directions
Name the opposite rays:
3)
2)
D
1)
A
B
BA and
48
H
H
C
G
L
K
I
E
J
F
BC
ED and
EF
EH and
EG
IL
and
IJ
Chapter 2, Section 8: “Vertical Angles”
After studying this SECTION, you should be able to . . .
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
Vertical Angles- (definition) – two angles whose sides are formed by opposite rays.
4) Which numbered angle
is vertical with ∡1?
5) Which numbered angle
is vertical with ∡4?
∡3
∡2
6) If m∡1 = 65, find the measure
of the numbered angles.
49
115°
D
H
2
65°
1
3
65°
4
G
F
Chapter 2, Section 8: “Vertical Angles”
After studying this SECTION, you should be able to . . .
2.8 Recognize opposite rays
2.8 Recognize Vertical Angles
Related Vocabulary
Vertical Angles- (definition) – two angles whose sides are formed by opposite rays.
7) If m∡3 = 55, which other
numbered angle must be 55°?
∡6
7) If m∡1 = 40, which other
numbered angle must be 40°?
∡4
50
40°
1
6
2
5
3
4
55°
Self-Check Properties Quiz Questions
1. Given:
A
E and D are the midpoints of
AC and AB,
E
and AC ≅ AB
D
F
Conclusion: CE ≅ DB
C
Because?
G
H
B
Addition
Subtraction
Multiplication
Division Like divisions of ≅ segs are ≅
Transitive
Substitution
51
Self-Check Properties Quiz Questions
2. Given:
A
FE ≅ FD, and
FC ≅ FB
E
D
Conclusion: CD ≅ EB
F
C
Because?
G
H
B
Addition
≅SEGS added to ≅SEGS are ≅ SEGS
Subtraction
Multiplication
Division
Transitive
Substitution
52
Self-Check Properties Quiz Questions
3. Given:
A
CD bisects ∡ACB,
BE bisects ∡ABC,
and ∡ACB ≅ ABC
E
D
F
C
Conclusion: ∡ACD ≅ ∡ABE
Because?
G
H
B
Addition
Subtraction
Multiplication
Like divisions of ≅ ∡s are ≅
Division
Transitive
Substitution
53
Self-Check Properties Quiz Questions
4. Given:
A
CG ≅ GH,
BH ≅ GH,
E
D
F
C
Conclusion: CG ≅ BH
Because?
G
H
B
Addition
Subtraction
Multiplication
Division
If segs are ≅ to SAME seg,
Transitive
then ≅ to each other
Substitution
54
Self-Check Properties Quiz Questions
5. Given:
A
∡BCD ≅ ∡CBE ,
and ∡ACB ≅ ABC
E
D
F
C
Conclusion: ∡ACD ≅ ∡ABE
Because?
G
H
B
Addition
Subtraction If ≅ ∡s are subtracted from ≅ ∡s,
then the like diffs are ≅
Multiplication
Division
Transitive
Substitution
55
Self-Check Properties Quiz Questions
6. Given:
A
EF = FD, and
EF + FB = EB
E
D
Conclusion: FD + FB = EB
F
=
Because?
!
C
G
H
B
Addition
Subtraction
Multiplication
Division
One seg measure can be
Transitive
Substitution substituted for the other in the
EQUATION!
56
Self-Check Properties Quiz Questions
7. Given:
A
CH ≅ BG
E
D
Conclusion: CG ≅ BH
F
C
Because?
G
H
B
Addition
Subtraction If the SAME seg is subtracted from
≅SEGS , the like diffs are ≅
Multiplication
Division
Transitive
Substitution
57
Self-Check Properties Quiz Questions
8. Given:
A
∡CAB + ∡ACB + ∡ABC = 180°
and ∡ACB ≅ ABC
E
D
F
C
G
H
B
Conclusion: 2(∡ABC) + ∡CAB = 180°
Because?
Addition
Subtraction
Multiplication
Division
One ANGLE measure can be
Transitive
Substitution substituted for the other in the
EQUATION!
58
Self-Check Properties Quiz Questions
9. Given:
A
CD bisects ∡ACB,
BE bisects ∡ABC,
and ∡ACD ≅ ABE
E
D
F
C
Conclusion: ∡ACB ≅ ∡ABC
Because?
Addition
Subtraction
Multiplication
Division
Transitive
Substitution
G
H
B
Like multiples of ≅ ∡s are ≅
59
Self-Check Properties Quiz Questions
10. Given:
A
∡AFD ≅ ∡AFE ,
and ∡DFB ≅ EFC
E
D
F
C
Conclusion: ∡AFC ≅ ∡AFB
Because?
Addition
Subtraction
Multiplication
Division
Transitive
Substitution
G
H
B
If ≅ ∡s are added to ≅ ∡s,
then the like sums are ≅
60