Download Chapter 1 Points, Lines, Planes, and Angles page 1

Lesson 2-3
Proving Theorems
(page 43)
Essential Question
Can you justify the conclusion
of a conditional statement?
Proving Theorems
What is a theorem?
Statements that can
be proved.
Midpoint Theorem
Theorem 2-1
If M is the midpoint of
AB,
½
½
then AM = ____ AB and MB = ____ AB.
Given:
M is the midpoint of AB
1
AB
Prove: AM =
2
1
MB = AB
2
⦁
A
⦁
M
⦁
B
Given:
M is the midpoint of AB
Prove:
AM =
1
AB
2
⦁
A
⦁
M
⦁
B
1
MB = AB
2
Statements
Reasons
M is the midpoint of AB
___________________
___________________
AM @ MB
___________________
AM = MB
___________________
AM + MB = AB
Given
Definition of Midpoint
___________________
___________________
Def. of Segments
Segment Add. Post.
___________________
5. ______________________
AM + AM = AB
Substitution Prop.
___________________
___________________
Distributive Prop.
___________________
Division Prop. of =
___________________
Substitution Prop.
1.
2.
3.
4.
6. ___________________
2 AM = AB
7. ___________________
AM = ½ AB
8. ___________________
MB = ½ AB
Example:
Given R is the midpoint of SQ .
Give the reason that justifies each statement.
S
R
P
(a)
Q
SR @ RQ
Definition of Midpoint
_________________________________
Example:
Given R is the midpoint of SQ .
Give the reason that justifies each statement.
S
R
P
(b)
Q
1
SR = SQ
2
Midpoint Theorem
_________________________________
Example:
Given R is the midpoint of SQ .
Give the reason that justifies each statement.
S
R
P
(c)
Q
SR + RQ = SQ
Segment Addition Post.
_________________________________
Example:
Given R is the midpoint of SQ .
Give the reason that justifies each statement.
S
R
P
(d)
Q
PR bisects SQ
Def. of Segment Bisector
_________________________________
Theorem 2-2
If
Angle Bisector Thm
BX is the bisector of ∠ABC,
½
and m∠XBC = ____
½ m∠ABC.
then m∠ABX = ____ m∠ABC
Given:
BX is the bisector of ÐABC
Prove:
1
mÐABX = mÐABC
2
1
mÐXBC = mÐABC
2
A
⦁
X
⦁
B
⦁
C
Given:
BX is the bisector of ÐABC
Prove:
mÐABX =
1
mÐABC
2
1
mÐXBC = mÐABC
2
Statements
A
⦁
X
⦁
B
Reasons
⦁
C
1. ___________________ ___________________
2. ___________________
___________________
See page
45
___________________
___________________
Classroom
3. ___________________ ___________________
Exercises
4. ______________________
___________________
#10 for HELP!
___________________
___________________
5. ___________________ ___________________
6. ___________________ ___________________
Given:
BX is the bisector of ÐABC
Prove:
mÐABX =
1
mÐABC
2
1
mÐXBC = mÐABC
2
Statements
A
⦁
X
⦁
B
Reasons
⦁
C
1. BX
___________________
is the bisector of ÐABC
2. ___________________
ÐABX @ ÐXBC
___________________
(m∠ABX
= m∠XBC)
m∠ABX + m∠XBC = m∠ABC
3. ___________________
___________________
Given
Def. of Angle Bisector
___________________
___________________
(Def. of Angles)
Angle Add. Post.
___________________
4. ______________________
m∠ABX + m∠ABX = m∠ABC
Substitution Prop.
___________________
(Distributive Prop.)
___________________
___________________
Division Prop. of =
___________________
Substitution Prop.
(___________________
2 m∠ABX = m∠ABC )
5. ___________________
m∠ABX = ½ m∠ABC
6. ___________________
m∠XBC = ½ m∠ABC
Example:
Given FD bisects ∠CFE.
Give the reason that justifies each statement.
C
D
F
(a)
E
m∠CFD = ½ m∠CFE
Angle Bisector Theorem
_________________________________
Example:
Given FD bisects ∠CFE.
Give the reason that justifies each statement.
C
D
F
(b)
E
m∠CFD = m∠DFE
Def. of Angle Bisector
_________________________________
Example:
Given FD bisects ∠CFE.
Give the reason that justifies each statement.
C
D
F
(c)
E
CD + DE = CE
Segment Addition Post.
_________________________________
Reason Used in Proofs
1. Given information
2. Definitions
3. Postulates (including properties
from algebra)
4. Theorems - only ones that have
already been proved!
Deductive Reasoning:
… proving statements by
reasoning from accepted
postulates, definitions,
theorems, and given
information.
Example:
2-column proofs
Note:
Definitions
can be written as
biconditionals
(combine conditional and converse),
i.e. the conditional and
converse are both true .
Example of a Biconditional:
• Conditional: If an angle is a right angle,
then its measure is 90º.
• Converse: If the measure of an angle is 90º,
then the angle is a right angle.
• Biconditional: An angle is a right angle if
and only if its measure is 90º.
Given: M is the midpt. of PQ
N is the midpt. of RS
PQ = RS
Prove: PM = RN
Statements
∙P
∙M
∙Q
∙R
∙N
∙S
Reasons
1. ___________________
___________________
Given
M is the midpt. of PQ
___________________
N is the midpt. of RS
___________________
PQ = RS
½ PQ = ½ RS
2. ______________________
Multiplication Prop.
___________________
PM = ½ PQ
3. ______________________
Midpoint Theorem
___________________
Midpoint Theorem
RN = ½ RS
4. ___________________
___________________
5. ___________________
___________________
Substitution Prop.
PM = RN
Assignment
Written Exercises on pages 46 & 47
RECOMMENDED: 9 to 12 ALL numbers, 19 & 20
REQUIRED: 1 to 8 ALL numbers, 13 & 14
Assignment
Worksheet on Lesson 2-3
Prepare for a Quiz on
Lessons 2-1 to 2-3: Using Deductive Reasoning
Can you justify the conclusion
of a conditional statement?