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Geometry 1
Unit 2: Reasoning and Proof
1
Geometry 1 Unit 2
2.1 Conditional Statements
2
Conditional Statements

Conditional StatementA

statement with two parts
If-then form
A
way of writing a conditional statement that clearly
showcases the hypothesis and conclusion

Hypothesis The

“if” part of a conditional Statement
Conclusion
 The
“then” part of a conditional Statement
3
Conditional Statements

Examples of Conditional Statements
 If
today is Saturday, then tomorrow is Sunday.
 If it’s a triangle, then it has a right angle.
 If x2 = 4, then x = 2.
 If you clean your room, then you can go to the
mall.
 If p, then q.
4
Conditional Statements


Example 1
Circle the hypothesis and underline the conclusion in
each conditional statement

If you are in Geometry 1, then you will learn about the building
blocks of geometry

If two points lie on the same line, then they are collinear

If a figure is a plane, then it is defined by 3 distinct points
5
Conditional Statements


Example 2
Rewrite each statement in if…then form
A
line contains at least two points
If a figure is a line, then it contains at least two points
 When
two planes intersect their intersection is a line
If two planes intersect, then their intersection is a
line.
 Two
angles that add to 90° are complementary
If two angles add to equal 90°, then they are
complementary.
6
Conditional Statements

Counterexample
 An
example that proves that a given
statement is false

Write a counterexample
 If x2
= 9, then x = 3
7
Conditional Statements

Example 3
 Determine
if the following statements are true
or false.
 If false, give a counterexample.
If x + 1 = 0, then x = -1
 If a polygon has six sides, then it is a decagon.
 If the angles are a linear pair, then the sum of the
measure of the angles is 90º.

8
Conditional Statements

Negation
 In
most cases you can form the negation of a
statement by either adding or deleting the
word “not”.
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Conditional Statements

Examples of Negations
 Statement:
mA  30
 Negation
:
mA  30
 Statement:
Mr. Ross is not more than 6 feet tall.
 Negation: Mr. Ross is more than 6 feet tall
 I am doing my homework.
 Negation:
10
Conditional Statements
Example 4
 Write the negation of each statement.
Determine whether your new statement is
true or false.


Stanfield is the largest city in Arizona.
 All triangles have three sides.
 Dairy cows are not purple.
 Some VGHS students have brown hair.
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Conditional Statements

Converse

Formed by switching the if and the then part.

Original


If you like green, then you will love my new shirt.
Converse

If you love my new shirt, then you like green.
12
Conditional Statements

Inverse
 Formed
by negating both the if and the then
part.

Original


If you like green, then you will love my new shirt.
Inverse

If you do not like green, then you will not love my new
shirt.
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Conditional Statements

Contrapositive
 Formed
by switching and negating both the if
and then part.

Original


If you like green, then you will love my new shirt.
Contrapositive

If you do not love my new shirt, then you do not like
green.
14
Conditional Statements
Write in if…then form.
 Write the converse, inverse and
contrapositive of each statement.

I
will wash the dishes, if you dry them.
A
square with side length 2 cm has an area of
4 cm2.
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Conditional Statements

Point-line postulate:
 Through
any two points, there exists exactly
one line

Point-line converse:
A

line contains at least two points
Intersecting lines postulate:
 If
two lines intersect, then their intersection is
exactly one point
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Conditional Statements

Point-plane postulate:
 Through
any three noncollinear points there exists
one plane

Point-plane converse:
A

plane contains at least three noncollinear points
Line-plane postulate:
 If
two points lie in a plane, then the line containing
them lies in the plane

Intersecting planes postulate:
 If
two planes intersect, then their intersection is a line
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Use the diagram to state the postulate that verifies the truth of the
statement.
The points E, R, and T lie in a plane (labeled A).
The points E and R lie on a line (labeled y).
The planes A and P intersect in a line (labeled m).
The points E and R lie in a plane A.
Therefore, line y lies in plane A.
m
P
●T
E
R
y
A
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Geometry 1 Unit 2
2.2: Definitions and Biconditional
Statements
19
Check your
answers on
Worksheet 2.1A
20
Do you remember…..
At the bottom of Page 5 of your notes packet, make a strip containing 10
boxes. In those boxes, write any of these terms. We are about to play….
collinear
perpendicular
congruent
vertical angles
inverse
supplementary
midpoint
adjacent angles
contrapositive
complementary
counterexample
coplanar
conditional statement
converse
21
Biconditional Statement
Two Statements Combined into One
I will pass if and only if I earn a 70% or better in this class.
I am happy if and only if I smile at Mrs. Dolezal.
I have good smelling breath if and only if I brush my teeth.
I attract more friends if and only if I learn geometry.
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Definitions and Biconditional Statements
Can be rewritten with “if and only if”
 Only occurs when the statement and the
converse of the statement are both true.
 A biconditional can be split into a
conditional and its converse.

23
Definitions and Biconditional Statements

Example 1 (Write as a conditional statement and its converse.)

An angle is right if and only if its measure is 90º

A number is even if and only if it is divisible by two.

A point on a segment is the midpoint of the segment if and only if it bisects the
segment.

You attend school if and only if it is a weekday.

You get an A if and only if you bring the teacher gifts.
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Definitions and Biconditional Statements

Perpendicular lines
 Two
lines are perpendicular if they intersect to form a
right angle

To the left of “Perpendicular Lines,” draw 5 lines that intersect.
Put your pen on the intersection so that it goes straight up from the intersection.

A line perpendicular to a plane
A
line that intersects the plane in a point and is
perpendicular to every line in the plane that intersects it

The symbol

is read, “is perpendicular to.
25
Definitions and Biconditional Statements

Example 2
 Write
the definition of perpendicular
as a biconditional statement.

_________________________ if and only if
_____________________________________
Definition of perpendicular: Perpendicular
objects form right angles at their intersection.
26
Definitions and Biconditional Statements

Example 3
 Give
a counterexample that demonstrates
that the converse is false.

(not on paper…) If you are in Mrs. Dolezal’s class,
you are having fun.

If two lines are perpendicular, then they intersect.
27
Definitions and Biconditional Statements

Example 4
 The
following statement is true. Write the
converse and decide if it is true or false. If the
converse is true, combine it with its original to
form a biconditional.

If x2 = 4, then x = 2 or x = -2
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Definitions and Biconditional Statements

Example 5
 Consider
the statement
x2 < 49 if and only if x < 7.
Is this a biconditional?
 Is the statement true?

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Person
Lower Limit
Upper Limit
Whiz Kid
90%
100%
Smarty
80%
89%
Average
70%
79%
Below Average
0%
69%
Use the information in the table to write a definition for each
type of student. For example: A student who has an average
of 90% to 100% is called a whiz kid.
30
WORKSHEET 2.2A
ANSWERS
31
Each person selects a slogan from the basket.
You and your table mate decides which slogan you use.
• Write the slogan as a conditional statement (if-then format).
• Write its converse, inverse, and contrapositive.
SLOGAN ACTIVITY
32
Geometry 1 Unit 2
2.3 Deductive Reasoning
33
Deductive Reasoning

Symbolic Logic
 Statements
are replaced with variables, such
as p, q, r.
 Symbols are used to connect the statements.
34
Deductive Reasoning
Symbol
~
→
Λ
V
→
↔
Meaning
not
if…then
and
or
if…then
if and only if
35
Deductive Reasoning

Example 1
 Let
p be “the sum of the measure of two
angles is 180º” and
 Let q be “two angles are supplementary”.

What does p → q mean?

What does q → p mean?
36
Deductive Reasoning

Example 2
 p:
Jen cleaned her room.
 q: Jen is going to the mall.
What does p → q mean?
 What does q → p mean?
 What does ~q mean?
 What does p Λ q mean?

37
Deductive Reasoning

Example 3


t: Jeff has a math test today
s: Jeff studied
tvs
 s → t
 ~s → t
 What does ~t mean?

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If Ana completes all her homework, then she will go to the movies.
Ana completed all of her homework. What will Ana do now?
If Joe wins the football game, he will get a new movie.
Joe did not win the football game. Will John get a new movie?
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If Derrick cleans his room, he will go to the mall.
If Derrick goes to the mall, he will get new shoes.
Derrick cleaned his room. Does he get new shoes?
LAW OF SYLLOGISM
40
Deductive Reasoning

Deductive Reasoning
 Deductive
reasoning uses facts, definitions,
and accepted properties in a logical order to
write a logical argument.
41
Deductive Reasoning

Law of Detachment
 When
you have a true conditional statement
and you know the hypothesis is true, you can
conclude the conclusion is true.
Given:
Given:
Conclusion:
p→q
p
q
42
Deductive Reasoning

Example 4
 Determine
if the argument is valid.
If Jasmyn studies then she will ace her test.
Jasmyn studied.
Jasmyn aced her test.
43
Deductive Reasoning

Example 5
 Determine
if the argument is valid.
If Mike goes to work, then he will get home late.
Mike got home late.
No Conclusion – Cannot make a valid argument.
44
Now Try These
If an angle is right, then it is not acute.
 TOP is a right angle.
Valid or Invalid: TOP is not acute.
If it is finger-licking good, then it is Kentucky Fried chicken.
It is Kentucky Fried chicken.
Valid or Invalid: It is finger-licking good.
If you use Pro-Active, then you will not have acne.
You don’t have acne.
Valid or Invalid: You use Pro-Active
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And more…
If you chew gum at VGHS, then you could get lunch detention.
You chew gum.
Valid or Invalid: You could get lunch detention.
If you go to Burger King, then you have it your way.
You go to Burger King.
Valid or Invalid: You have it your way.
If you have an LG television, then life is good.
Life is good.
Valid or Invalid: You have an LG television.
If you want to reach out and touch someone, then use AT & T.
You want to reach out and touch someone.
Valid or Invalid: You use AT & T.
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WHY ARE FIRETRUCKS RED?
Cause there's eight wheels on them and four people, and four plus
eight is twelve, and twelve is a foot and a foot is a ruler, and Queen
Elizabeth was a ruler, and Queen Elizabeth was also a ship, and
the ship sails the sea and in the sea is fish and fish have fins, and
the Finns fought the Russians and the Russians were red and
that's why firetrucks are red.
47
Use the Law of Syllogism to complete the statement,
”If there is a fire, then __________________.”
If the robot sets off a fire alarm, then it concludes there is a fire.
If the robot senses high levels of smoke and heat, then it sets off a fire alarm.
If the robot locates the fire, then the robot extinguishes the fire.
If there is a fire, then the robot senses high levels of smoke and heat.
If the robot concludes there is a fire, then it locates the fire.
48
Use the Law of Syllogism to complete the statement,
“If an Old Lady swallowed a bat, then _______________.”
From There was an Old Woman Who Swallowed a Bat by Lucille Colandro.
If she swallowed an owl, then she swallowed a cat.
If she swallowed a ghost, then she swallowed a goblin.
If an Old Lady swallowed a bat, then she swallowed an owl.
If she swallowed a wizard, then she yelled, “Trick of treat!”
If she swallowed a cat, then she swallowed a ghost.
If she swallowed a goblin, then she swallowed some bones.
If she swallowed some bones, then she swallowed a wizard.
49
A nursery rhyme ends with
“For want of a nail, the _________.”
If the horseshoe is lost, then the horse will be lost.
If the horse is lost, then the knight will be lost.
If the horseshoe nail is lost, then the horseshoe will be lost.
If the battle is lost, then the kingdom will be lost.
If the knight is lost, then the battle will be lost.
50
Take out a sheet of paper.
Write an if-then statement on your paper. It can be about anything.
For example….If I forgot my homework today, then I get a 0.
Pass the paper to the person at the desk numbered one
higher than yours.
Using the conclusion of the last sentence as your hypothesis, write
another conditional.
For example…If I got a 0, then my grade will drop.
Pass your paper to the next person again.
AN EXERCISE IN SYLLOGISM
51
Deductive Reasoning

Law of Syllogism
 Given
two linked conditional statements you
can form one conditional statement.
Given:
Given:
Conclusion:
p→q
q→r
p→r
52
Deductive Reasoning

Example 6
 Determine
if the argument is valid.
If today is your birthday, then Joe will bake a cake.
If Joe bakes a cake, then everyone will celebrate.
If today is your birthday, then everyone will celebrate.
53
Deductive Reasoning

Example 7
 Determine
if the argument is valid.
If it is a square, then it has four sides.
If it has four sides, then it is a quadrilateral.
If it is a square, then it is a quadrilateral.
54
Now Try These
Write a conclusion using the true statements. If no
conclusion is possible, write no conclusion.
If Jesse is late, then he is tardy.
If he is tardy, then he will get lunch detention.
Jesse is tardy….
If Casey is friendly, then she will have a date.
If she has a date, then she will go to up-and-coming.
Casey is going to up-and-coming.
If Mary goes to Vista Grande, then she is a Spartan.
If she is a Spartan, then she has school pride.
Mary is a Spartan.
55
More…
Write a conclusion using the true statements. If no
conclusion is possible, write no conclusion.
If Tim misses practice, then he cannot play in the game.
Tim goes to practice.
If Deb does her homework, then it will be graded.
If the homework is graded, then it will help her pass.
Deb did her homework.
If Sara attends class every day, then she will have perfect attendance.
If she has perfect attendance, then she will not have to take finals.
If she does not have to take finals, then she will have get out of school 2
days early.
Sara attends class every day.
56
If you are tall, then you play
basketball.
You are tall.
You play basketball.
You are not tall.
You do not play basketball.
If you are late, then you get
lunch detention.
You are tardy.
You have lunch detention.
You are not tardy.
You do not have lunch detention.
ANOTHER WAY TO DETERMINE
VALIDITY USING VENN DIAGRAMS
57
Geometry 1 Unit 2
2.4 Reasoning with Properties
from Algebra
58
Reasoning with
Properties from Algebra

Addition property
 If

Subtraction property
 If

a = b, then a – c = b – c
Multiplication property
 If

a = b, then a + c = b + c
a = b, then ac = bc
Division property
 If
a = b, then
a c  b c
59
Reasoning with
Properties from Algebra

Reflexive property
 For

Symmetric property
 If

a=b, then b = a
Transitive Property
 If

any real number a, a = a
a = b and b = c, then a = c
Substitution property
 If
a = b, then a can be substituted for b in any
equation or expression

Distributive property
 2(x
+ y) = 2x + 2y
60
USING THE PROPERTY IN LIFE
Property
Example
Addition Property
Allowance
Reflexive Property
I am who I am.
Substitution Property
Cooking Ingredients
Transitive Property
If I’m as good as you and you’re as good as ….., then…
Symmetric Property
Turning About Face
Multiplication Property
Wages on an Hourly Basis
Subtraction Property
Payments
Division Property
Sharing
Distributive Property
Passing Out Rewards
61
USING THE PROPERTY IN GEOMETRY
Property
Example
Addition Property
If mTRY = 90°, then 10 + mTRY = _____
Reflexive Property
M = _____
Substitution Property
If mH = 120°, then mH + mI = 120 + mI
Transitive Property
If GR = EA and ____ = TM, then _____________.
Symmetric Property
CAT + 50 = 50 + CAT
Multiplication Property
If mTRY = 90°, 2(mTRY) = _____
Subtraction Property
If mNOT = mTRY, then mNOT - LOW = mTRY - LOW.
Division Property
If mLOW = mTRY, then (mLOW)/2 = (mTRY)/2
Distributive Property
If 2MAD +2 TRY = 90, then 2(MAD + TRY) = 90.
62
CAN YOU IDENTIFY THE PROPERTY?
1. Cut into strips and cut properties from examples.
2. Match the property to the example.
Property
Example
Addition Property
If mK = 10°, then 4(mK) = 40°
Reflexive Property
If mM = mA and mA = mT, then mM = mT.
Substitution Property
If 8(mV) = 120°, then mV = 15°
Transitive Property
If AB = 10 cm, then AB + 5 cm = 15 cm.
Symmetric Property
CAT = CAT
Multiplication Property
If TH = UM, then UM = TH
Subtraction Property
If TRY = 90°, then MIS + TRY = MIS + 90.
Division Property
If LOW + TRY = HAM + TRY, then LOW = HAM
Distributive Property
If 2(LOW + TRY) = 90, then 2LOW +2 TRY = 90.
63
Reasoning with
Properties from Algebra

Example 1
 Solve
6x – 5 = 2x + 3 and write a reason for each step
Statement
6x – 5 = 2x + 3
Reason
Given
4x – 5 = 3
4x = 8
x=2
64
Reasoning with
Properties from Algebra
Example 2
 2(x – 3) = 6x + 6
1.

2.

3.

4.

5.
Given
65
Create Your Own
Step
Reason
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
66
Another Try (but not on notes)
Step
Reason
1. 3x - 7= 5(2 + x) + 5
1.
2. 3x - 7 = 10 + 5x + 5
2.
3. 3x - 7 = 15 + 5x
3.
4. -22 = 2x
4.
5.
5.
2x = - 22
6. x = - 11
6.
67
Still Another Try (but not on notes)
Step
Reason
1. x = 4 + y
3x + 7(y + 3) = 53
1.
2. 3(4 + y) + 7(y + 3) = 53
2.
3. 12 + 3y + 7y + 21 = 53
3.
4. 33 + 10y = 53
4.
5.
5.
10y = 20
6. y = 2
6.
7. x = 4 + y
7.
8. x = 4 + 2
8.
9. x = 6
9.
68
Reasoning with
Properties from Algebra

Determine if the equations are valid or invalid.
 (x
+ 2)(x + 2) = x2 + 4
 x3x3
 -(x
= x6
+ y) = x – y
69
More Reasoning with
Properties from Algebra

(not on notes)
Determine if the equations are valid or invalid.
 (x
+ 2)(x - 2) = x2 - 4
 (x3)3
= x6
 -5(x
- y) = -5x + y
70
Reasoning with
Properties from Algebra

G
Geometric Properties of Equality
 Reflexive

property of equality
For any segment AB, AB = AB
 Symmetric
property of equality
 If mA  mB, then mB  mA
 Transitive property of equality

B
A
P
If AB = CD and CD = EF, then, AB = EF
71
Reasoning with
Properties from Algebra
Example 3
A
B
C
D
In the diagram, AB = CD. Show that AC = BD
Statement
Reason
1. AB = CD
2. AB + BC = BC + CD
3. AC = AB + BC
4. BD = BC + CD
5. AC = BD
72
A
In the diagram, ABC = DBF.
Show that  ABD = CBF
C
D
B
F
Statement
Reason
73
Reasoning with
Properties from Algebra
Example 4
A
(not on notes)
B
C
D
In the diagram, AC = DB. Show that AB = CD
Statement
Reason
1. AC = DB
2. BC = BC
3. AB + BC = AC
4. CD + BC = DB
5. AB + BC = CD + BC
6. AB = CD
74
A
In the diagram, ABD = CBF.
Show that  ABC = DBF
C
Not on notes.
D
B
F
Statement
Reason
75
Geometry 1 Unit 2
2.5: Proving Statements about
Segments
76
Marking Diagrams
LN  MP
77
Proving Statements about
Segments

Key Terms:
 2-column

proof
A way of proving a statement using a numbered
column of statements and a numbered column of
reasons for the statements
 Theorem

A true statement that is proven by other true
statements
78
Proving Statements about
Segments

Properties of Segment Congruence
 Reflexive

For any segment AB, AB  AB
 Symmetric

If AB  CD, then CD  AB
 Transitive

If AB  EF and AB  CD ,then CD  EF
79
Proving Statements about
Segments

Example 1
K
 In
triangle JKL,
Given: LK = 5, JK = 5, JK = JL
Prove: LK = JL
J
L
Statement
1.
Reason
1. Given
2.
3.
4.
2. Given
3. Substitution
4. Transitive Property of Congruence
80
Proving Statements about
Segments (not on notes)

Example 2

L
Given: M is the midpoint of LN
Prove: LM = ½ LN and MN = ½ LN
Statement
Reason
1.
1. Given
2. LM = MN
2.
3. LM + MN = LN
3.
4. LM + LM = LN
4.
5. 2 * LM = LN
5.
6. LM = ½ LN
6.
7.
7.
M
N
81
Proving Statements about
Segments (not on notes)

Example 3

W
X
Given: Collinear Points W, X, Y and Z
Prove: WZ = WX + XY + YZ
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
Y
Z
82
Proving Statements about
Segments (not on notes)

Example 5

M A
T
Given: Collinear Points M, A, T, and H
Prove: MH = MA + AT + TH
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
H
83
Proving Statements about
Segments (not on notes)

Example 6
G
R

M
B
GR  EA
Prove: GR  BC
 Given:
EA  TM
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
C
TM  MN
N
T M
MN  BC
84
Proving Statements about
Segments


Duplicating a Segment
Tools


Straight edge: Ruler or piece
of wood or metal used for
creating straight lines
 Compass: Tool used to create
arcs and circles
Steps
1.
2.
3.
A
C
B
D
4.
5.
6.
Use a straight edge to
draw a segment longer
than segment AB
Label point C on new
segment
Set compass at length of
segment AB
Place compass point at C
and strike an arc on line
segment
Label intersection of arc
and segment point D
Segment CD is now
congruent to segment AB85
NOT ON NOTES
A
AB
B
C
D
AB – CD
CD
3AB – 2CD
AB + CD
86
A
2AB
B
C
D
3CD - AB
87
Let’s Try More Constructions
Bisect an Angle
Bisect a Segment
88
Geometry 1 Unit 2
2.6: Proving Statements about
Angles
89
Proving Statements about Angles

Properties of Angle Congruence
 Reflexive
For any angle A, A  A.
 Symmetric
If A  B, thenB  A.
 Transitive
If A  Band B  C , thenA  C.
90
Proving Statements about Angles

Right Angle Congruence Theorem
 All
right angles are congruent.
91
Proving Statements about Angles

Congruent Supplements Theorem
 If
two angles are supplementary to the same angle,
then they are congruent.
If
m1  m2  180 and
m2  m3  180 ,
then1  3.
1
2
3
92
Visual of Supplementary Angles
(not on notes)
Sketch
supplementary
angles where one
angle measures
135°.
Sketch
supplementary
angles where one
angle measures 30°.
93
Proving Statements about Angles

Congruent Complements Theorem
 If
two angles are complementary to the same angle,
then the two angles are congruent.
If
m4  m5  90 and
m5  m6  90 ,
then4  6.
5
6
4
94
Visual of Complementary Angles (not on notes)
Sketch
complementary
angles where one
angle measures 60°.
Sketch
complementary
angles where one
angle measures 35°.
95
Proving Statements about Angles

Linear Pair Postulate
 If
two angles form a linear pair, then they are
supplementary.
m1  m2  180
1
2
96
Visual of Linear Pair (not on notes)
Sketch a linear pair
where one angle
measures 35°.
Sketch a linear pair
where one angle
measures 135°.
97
Proving Statements about Angles

Vertical Angles Theorem
 Vertical
angles are congruent.
2
1
3
4
1  3, 2  4
98
Visual of Vertical Angles (not on notes)
Sketch vertical
angles which
measure 40°.
Sketch vertical
angles which
measure 110°.
99
Proving Statements about Angles

Example 1
 Given:1  2, 3  4, 2  3.
B
 Prove:1  4
1
A
Statement
1.
2.
3.
4.
Reason
1.
2.
3.
4.
3
2
4
C
100
Proving Statements about Angles

Example 2
 Given: m1  63 , 1  3, 3  4
 Prove:
m4  63
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
1
3
2
4
101
Proving Statements about Angles
D

Example 3
A
 Given: DAB, ABC are right angles
C
B
ABC  BDC
 Prove:
DAB  BDC
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
102
Proving Statements about Angles

Example 4


Given:
m1 = 24º,
m3 = 24º
1 and 2 are
complementary
3 and 4 are
complementary
Prove: 2  4
1 2
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
3
4
103
Proving Statements about Angles

Example 5
the diagram m1 = 60º and BFD is right.
Explain how to show m4 = 30º.
 In
C
B
D
1
A
2 3
F
4
E
104
Proving Statements about Angles

Example 6
1 and 2 are
a linear pair, 2 and
3 are a linear pair
 Prove: 1  3
 Given:
1
2
3
Statement
Reason
1.
1.
2.
2.
3.
3.
105
Write a two-column proof
Given: 8 = 5
Prove: 7 = 6
6
5
8
7
106