Identify the hypothesis and conclusion of each conditional; write the converse of the conditional; decide whether the conditional and converse are true or false. If false, provide a counterexample.

1. If you attend CB West, then you are in high school.

2. If 4x = 20, then x = 5.

1.If you attend CB West, then you are in high school.

Hypothesis: You attend CB West.

Converse: If you are in high school, then you attend CB West.

Conclusion: You are in high school.

False

Counterexample: You could attend CB South.

2. If 4x = 20, then x = 5.

Hypothesis: 4x = 20.

Converse: If x = 5, then 4x = 20. Tru

e

Conclusion: x = 5

1) Hypothesis: 2x – 1 = 5 Conclusion: x = 3

2) Hypothesis: She’s smart Conclusion: I’m a genius

3) Hypothesis: 8y = 40 Conclusion: y = 5

4) Hypothesis: S is the midpoint of RT Conclusion: RS = ½RT

5) Hypothesis: m1 = m2 Conclusion: 1 2

6) Hypothesis: 1 2 Conclusion: m1 = m2

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: m1 = 90 Conclusion: 1 is a right

angle5) Hypothesis: a + b = a

Conclusion: b = 06) Hypothesis: x = -5

Conclusion: x² = 25

7) B is between A and C if and only if AB + BC = AC

8) mAOC = 180 if and only if AOC is a straight angle.

Properties from Equality

Addition Property: If a = b, then a + c = b + c.

Subtraction Property: If a = b, then a - c = b - c.

Multiplication Property: If a = b, then ca = cb.

Division Property: If a = b and c 0, then c

b

c

a

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

Transitive Property: --If D B and B C, then D

C.

Introduction to Proofs

Statements

Reasons

All reasons must be:

1. “Given”2. Postulates3. Theorems4. Definitions5. Properties

Given:

Prove:

What you know

What you’re trying to prove

All of your math steps go here.

Example 1: Given: 3x – 10 = 20 Prove: x = 10.

Statements Reasons

1. 3x – 10 = 20

2. 3x – 10 + 10 = 20 + 10 3x = 30

10x 3

303x3

.3

1. Given

2. Addition Property

3. Division Property

Example 2: Given: 3x + y = 22; y = 4 Prove: x = 6

Statements Reasons

1. 3x + y = 22; y = 4

2. 3x + 4 = 22

1. Given

2. Substitution

4. Division Property

3. 3x = 18 3. Subtraction Property

4. x = 6

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 mAQB + mBQC = mAQC.

IfAQC is a straight angle and B is a point not on AC, then mAQB + mBQC =

180.A

Q C

B

A Q C

B

If you used this statement above, it would also be acceptable to use “definition of a linear pair” as

your reason.

Statements

Reasons

Given: FL = ATProve: FA = LT

F L A T

1. Given

2. Reflexive Property

3. Addition Property

5. Substitution

Example 3:

4. Segment Addition Postulate

1. FL = AT

2. LA = LA

3.FL + LA = AT + LA

4. FL + LA = FA LA + AT = LT

5. FA = LT

Sometimes, this step is not necessary. You could just go right to the next step.

Example 4:

Given: mAOC = mBOD Prove: m1 = m3

Statements

Reasons

1. mAOC = mBOD

2.mAOC = m1 + m2 mBOD = m2 + m3

3.m1 + m2 = m2 + m3

4. m2 = m2

5. m1 = m3

1. Given

2. Angle Addition Postulate

3. Substitution

5. Subtraction Property

4. Reflexive

A

O

B C

D321

Sometimes, this step is not necessary. You could just go right to the next step.

Statements Reasons

Example 5:

1. RS = PS 1. Given

2. RS + ST = PS + ST

2. Addition Property

5. RS + ST = RT PS + SQ = PQ

5. Segment Addition Postulate6. RT = PQ

4. Substitution

RP

Q T

S

4. RS + ST = PS + SQ

3. ST = SQ 3. Given

6. Substitution

Given: RS = PS and ST = SQ.Prove: RT = PQ