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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.
Theorem 2-1
Midpoint Theorem
If M is the midpoint of
AB,
½
½
then AM = ____ AB and MB = ____ AB.
Given:
M is the midpoint of AB
Prove:
1
AM =
AB
2
1
MB = AB A
2
M
B
Given:
M is the midpoint of AB
Prove:
AM =
1
AB
2
A
M
B
1
MB = AB
2
Statements
Reasons
1. ___________________ ___________________
2. ___________________ ___________________
See
page
43
3. ___________________ ___________________
for the proof.
4. ______________________
___________________
5. ___________________ ___________________
6. ___________________ ___________________
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
BX is the bisector of ÐABC
1
Prove:
mÐABX = mÐABC
2
1
B
mÐXBC = mÐABC
2
Given:
A
⦁
X
⦁
C⦁
Given:
BX is the bisector of ÐABC
Prove:
mÐABX =
1
mÐABC
2
1
mÐXBC = mÐABC
2
Statements
A
⦁
X
⦁
B
Reasons
⦁
C
1. ___________________ ___________________
See
page
45
2. ___________________ ___________________
Classroom
___________________
___________________
Exercises
3. ___________________
___________________
4. ______________________
#10.___________________
___________________ ___________________
5. ___________________ ___________________
6. ___________________ ___________________
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 page 46
DO NOW: 9 to 12 ALL numbers
GRADED: 1 to 8 ALL numbers
Can you justify the conclusion
of a conditional statement?
Prepare for a Quiz on
Lessons 2-1 to 2-3: Using Deductive Reasoning
Assignment
Worksheet on Lesson 2-3