Introduction to Proof

The student is able to (I can):

Prove geometric theorems by deductive reasoning

Use angles formed by a transversal to show that lines are Use angles formed by a transversal to show that lines are parallel.

Properties of Equality

One of the basic principles of algebra is that whatever you do to one side of an equation, you must do the same thing to the other. This idea actually comes from what are called properties of equality

Addition Property of Equality

Subtraction Property of Equality Subtraction Property of Equality

Multiplication Property of Equality

Division Property of Equality

Reflexive Property of Equality

Symmetric Property of Equality

Transitive Property of Equality

Substitution Property of Equality

add. prop. =

subtr. prop. =

Addition Property of Equality

If a = b, then a + c = b + c

Ex: If x 2 = 4, then we can add 2 to both sides, which means that x = 6.

Subtraction Property of Equality

If a = b, then a c = b c

Ex: If x + 7 = 12, then we can subtract 7

mult. prop. =

Ex: If x + 7 = 12, then we can subtract 7 from both sides, which means that x = 5.

Multiplication Property of Equality

If a = b, then ac = bc

Ex: If then we can multiply both sides by 2, which means that x = 20.

=1

2x 10,

div. prop. =

refl. prop. =

Division Property of Equality

If a = b and c 0, then

Ex. If 3x = 12, then we can divide both sides by 3, which means x = 4.

Reflexive Property of Equality

a = a

a b.

c c=

sym. prop. = Symmetric Property of Equality

If a = b, then b = a.

trans. prop. =

subst. prop. =

Transitive Property of Equality

If a = b, and b = c, then a = c.

Ex: If 1 dime = 2 nickels, and 2 nickels = 10 pennies, then 1 dime = 10 pennies.

Substitution Property of Equality

If a = b, then b can be substituted for a in any expression.

dist. prop.

in any expression.

Ex: If x = 5, then we can substitute 5 for x in the equation y = x + 2, which means y = 7.

We will also use the Distributive Property:

a(b + c) = ab + ac and ab + ac = a(b + c)

Line segments with equal lengths are congruent, and angles with equal measures are also congruent. Therefore, the reflexive, symmetric, and transitive properties of equality have corresponding properties of properties of properties of properties of congruencecongruencecongruencecongruence.

Reflexive Property of Congruence

fig. A fig. A

Symmetric Property of Congruence

If fig. A fig. B, then fig. B fig. A.

Transitive Property of Congruence

If fig. A fig. B and fig. B fig. C, then fig. A fig. C.

geometric proof

A proof which uses geometric properties and definitions

A two-column geometric proof begins with the GivenGivenGivenGiven statement and ends with the ProveProveProveProve statement.

List the steps of the proof in the left column and the justifications (reasons) column and the justifications (reasons) in the right column.

You may use definitions, postulates, and previously proven theorems as reasons.

Other types of proofs are

Paragraph proofs

Flowchart proofs

GivenGivenGivenGiven:::: BAC is a right angle;

2 3

ProveProveProveProve:::: 1 and 3 are complementary

12

3

B

A C

StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons

1. BAC is a right angle 1. Given

2. mBAC = 90 2. _______________2. mBAC = 90 2. _______________

3. _______________________ 3. Add. post.

4. m1 + m2 = 90 4. Subst. prop. =

5. 2 3 5. Given

6. _______________________ 6. Def. s

7. m1 + m3 = 90 7. _______________

8. _______________________ 8. Def. comp. s

GivenGivenGivenGiven:::: BAC is a right angle;

2 3

ProveProveProveProve:::: 1 and 3 are complementary

12

3

B

A C

StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons

1. BAC is a right angle 1. Given

2. mBAC = 90 2. _______________Def. right Def. right Def. right Def. right 2. mBAC = 90 2. _______________

3. _______________________ 3. Add. post.

4. m1 + m2 = 90 4. Subst. prop. =

5. 2 3 5. Given

6. _______________________ 6. Def. s

7. m1 + m3 = 90 7. _______________

8. _______________________ 8. Def. comp. s

Def. right Def. right Def. right Def. right

mmmm1 + m1 + m1 + m1 + m2 = m2 = m2 = m2 = mBACBACBACBAC

Subst. prop. =Subst. prop. =Subst. prop. =Subst. prop. =

mmmm2 = m2 = m2 = m2 = m3333

1 and 1 and 1 and 1 and 3 are comp.3 are comp.3 are comp.3 are comp.

GivenGivenGivenGiven 1 and 2 are supplementary, and

2 and 3 are supplementary

Prove Prove Prove Prove 1 3

1. 1 and 2 are supp. 1. Given

2 and 3 are supp.

2. m1 + m2 = 180 2. Def. supp. 2. m1 + m2 = 180 2. Def. supp. m2 + m3 = 180

3. 180 = m2 + m3 3. Sym. prop =

4. m1+m2=m2+m3 4. Trans. prop =

5. m1 = m3 5. Subtr. prop.=

6. 1 3 6. Def. s

Proving Lines Parallel

Recall that the converseconverseconverseconverse of a theorem is found by exchanging the hypothesis and conclusion. The converses of the parallel line theorems can be used to prove lines parallel.

Corresponding AnglesIf angles are congruent,

Alternate Interior Angles

Alternate Exterior Angles

Same-Side Interior Angles

If angles are supplementary, then the lines are parallel.

If angles are congruent, then the lines are parallel.

Example Find values of x and y that make the red lines parallel and the blue lines parallel.

(x40) (x+40)

y

If the blue lines are parallel, then the same-side interior angles must be supplementary.

x 40 + x + 40 = 180

2x = 180

x = 90

Example Find values of x and y that make the red lines parallel and the blue lines parallel.

(x40) (x+40)

y

If the red lines are parallel, then the same-side interior angles must be supplementary.

90 40 + y = 180

50 + y = 180

y = 130

Example: Complete the proof below.

Given: Given: Given: Given: 1 2

Prove:Prove:Prove:Prove: k || m k

m

1

2

StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons

1. 1 2 1.

2. k || m 2.

Example: Complete the proof below.

Given: Given: Given: Given: 1 2

Prove:Prove:Prove:Prove: k || m k

m

1

2

(Converse of the corresponding angles theorem)

StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons

1. 1 2 1. GivenGivenGivenGiven

2. k || m 2. Conv. corr.Conv. corr.Conv. corr.Conv. corr. ssss