Download 2.2 Intro to Proofs

Identify the hypothesis and conclusion of each
conditional; write the converse of the
conditional; decide whether the conditional
and converse are true or false.
1.
If you attend CB West, then you are in
high school.
2.
If 4x = 20, then x = 5.
3.
If tomorrow is Thursday, then today is
Wednesday.
1.If you attend CB West, then you are
in high school.
Hypothesis: You attend CB West.
Conclusion: You are in high school.
Converse: If you are in high school,
then you attend CB West. False
Counterexample: If you are in high school,
then you could go to CB East.
2. If 4x = 20, then x = 5.
Hypothesis: 4x = 20.
Conclusion: x = 5
Converse: If x = 5, then 4x = 20.
True
3. If tomorrow is Thursday,
then today is Wednesday.
Hypothesis: Tomorrow is Thursday.
Conclusion: Today is Wednesday.
Converse: If today is Wednesday,
then tomorrow is Thursday.
True
1) Hypothesis:
Conclusion:
2) Hypothesis:
Conclusion:
3) Hypothesis:
Conclusion:
4) Hypothesis:
Conclusion:
5) Hypothesis:
Conclusion:
6) Hypothesis:
Conclusion:
2x – 1 = 5
x=3
She’s smart
I’m a genius
8y = 40
y=5
S is the midpoint of RT
RS = ½RT
m1 = m2
1  2
1  2
m1 = m2
1) Hypothesis: 3x – 7 = 32
Conclusion: x = 13
2) Hypothesis: I’m not tired
Conclusion: I can’t sleep
3) Hypothesis: You will
Conclusion: I’ll try
4) Hypothesis: m1 = 90
Conclusion: 1 is a right
angle
5) Hypothesis: a + b = a
Conclusion: b = 0
6) Hypothesis: x = -5
Conclusion: x² = 25
7) B is between A and
C if and only if AB
+ BC = AC
8) mAOC = 180 if
and only if AOC
is a straight angle.
Warm Up - Solve these two problems for
x; show all work.
4x + 12 = 36
5x + 2y = 33; y = 4
Solve these two problems; show all work.
4x + 12 = 36
x=6
5x + 2y = 33; y = 4
x=5
ORDER IS IMPORTANT.
You need to always know
what information you have
and then draw conclusions
properly. For definitions,
order does not matter.
Properties from Equality
Addition Property: If a = b and c = d,
then a + c = b + d.
Subtraction Property: If a = b and c = d,
then a - c = b - d.
Multiplication Property: If a = b, then
ca = cb.
Division Property:
a b

If a = b and c  0, then c c
Properties of Equality
Substitution Property: if a = b, then
either a or b may be substituted for the
other in any equation or inequality.
Reflexive Property: a = a.
Symmetric Property: if a = b, then b = a.
Transitive Property: if a = b and b = c,
then a = c.
Properties of Congruence
Reflexive Property: DE  DE ; D  D
Symmetric Property:
--If DE  FG, then FG  DE;
--If D  B, then B  D.
Transitive Property:
--If DE  FG and FG  JK, then DE  JK;
--If D  B and B  C, then D  C.
Introduction to Proofs
Given: What you know
Prove: What you’re trying to prove
Statements
Reasons
All reasons must be:
1. Given
2. Postulates
3. Theorems
4. Definitions
5. Properties
Example 1:
Given: 3x – 10 = 20
Prove: x = 10.
Statements
Reasons
1. 3x – 10 = 20
1.
2. 3x – 10 + 10 = 20 + 10
3x = 30
2. Addition Property
3x 30
3.

3
3
x  10
Given
3. Division Property
Example 2:
Given: 3x + y = 22; y = 4
Prove: x = 6
Statements
Reasons
1. 3x + y = 22; y = 4
1.
2. 3x + 4 = 22
2.
3. 3x = 18
3. Subtraction Property
4. x = 6
4. Division Property
Given
Substitution
Because you use postulates, properties,
definitions, and theorems for your reasons in
proofs, it is a good idea to review the
vocabulary and postulates that we learned in
Unit 1.
Segment Addition Postulate –
If point B is between points A and C, then
AB + BC = AC.
A
B
C
Angle Addition Postulate –
If point B lies in the interior of AQC, then
mAQB + mBQC = mAQC.
If AQC is a straight angle and B is a point
not on AC, then mAQB + mBQC = 180.
B
B
A
Q
C
A
Q
C
If you used this statement above, it would also be
acceptable to use “definition of a linear pair” as your
reason.
Example 3:
Given: FL = AT
Prove: FA = LT
F
L
A
Statements
Reasons
1.
FL = AT
1.
Given
2.
LA = LA
2.
Reflexive Property
3.FL + LA = AT + LA
3.
Addition Property
4. FL + LA = FA
LA + AT = LT
4.
Segment Addition
Postulate
5.
5.
Substitution
FA = LT
T
Example 4:
B
A
Given: mAOC = mBOD
Prove: m1 = m3
Statements
C
1 2
3
O
Reasons
1. mAOC = mBOD
1.
2.mAOC = m1 + m2
mBOD = m2 + m3
2. Angle Addition
Postulate
3.m1 + m2 = m2 + m3
3.
Substitution
4.
4.
Reflexive
5. m1
m2 = m2
=
m3
Given
5. Subtraction
Property
D
Example 5:
Given: RS = PS and ST = SQ.
Prove: RT = PQ
Statements
1.
RS = PS
2. RS + ST = PS + ST
3.
ST = SQ
4. RS + ST = PS + SQ
5. RS + ST = RT
PS + SQ = PQ
6.
RT = PQ
R
S
Q
Reasons
1.
Given
2. Addition Property
3.
Given
4. Substitution
5. Segment Addition
Postulate
6. Substitution
P
T