* Prove geometric theorems by deductive reasoning * Use angles formed by a transversal to show that lines are parallel.

1. 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.

2. 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

3. 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
2
x 10,

4. 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.

5. 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)

6. 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 ofproperties ofproperties ofproperties 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.

7. 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

8. GivenGivenGivenGiven:::: ∠BAC is a right angle;
∠2 ≅ ∠3
ProveProveProveProve:::: ∠1 and ∠3 are complementary
1
2
3
•
•
B
A C
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. ∠BAC is a right angle 1. Given
2. m∠BAC = 90° 2. _______________2. m∠BAC = 90° 2. _______________
3. _______________________ 3. ∠ Add. post.
4. m∠1 + m∠2 = 90° 4. Subst. prop. =
5. ∠2 ≅ ∠3 5. Given
6. _______________________ 6. Def. ≅ ∠s
7. m∠1 + m∠3 = 90° 7. _______________
8. _______________________ 8. Def. comp. ∠s

9. GivenGivenGivenGiven:::: ∠BAC is a right angle;
∠2 ≅ ∠3
ProveProveProveProve:::: ∠1 and ∠3 are complementary
1
2
3
•
•
B
A C
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. ∠BAC is a right angle 1. Given
2. m∠BAC = 90° 2. _______________Def. rightDef. rightDef. rightDef. right ∠∠∠∠2. m∠BAC = 90° 2. _______________
3. _______________________ 3. ∠ Add. post.
4. m∠1 + m∠2 = 90° 4. Subst. prop. =
5. ∠2 ≅ ∠3 5. Given
6. _______________________ 6. Def. ≅ ∠s
7. m∠1 + m∠3 = 90° 7. _______________
8. _______________________ 8. Def. comp. ∠s
Def. rightDef. rightDef. rightDef. right ∠∠∠∠
mmmm∠∠∠∠1 + m1 + m1 + m1 + m∠∠∠∠2 = m2 = m2 = m2 = m∠∠∠∠BACBACBACBAC
Subst. prop. =Subst. prop. =Subst. prop. =Subst. prop. =
mmmm∠∠∠∠2 = m2 = m2 = m2 = m∠∠∠∠3333
∠∠∠∠1 and1 and1 and1 and ∠∠∠∠3 are comp.3 are comp.3 are comp.3 are comp.

10. GivenGivenGivenGiven ∠1 and ∠2 are supplementary, and
∠2 and ∠3 are supplementary
ProveProveProveProve ∠1 ≅ ∠3
1. ∠1 and ∠2 are supp. 1. Given
∠2 and ∠3 are supp.
2. m∠1 + m∠2 = 180° 2. Def. supp. ∠2. m∠1 + m∠2 = 180° 2. Def. supp. ∠
m∠2 + m∠3 = 180°
3. 180° = m∠2 + m∠3 3. Sym. prop =
4. m∠1+m∠2=m∠2+m∠3 4. Trans. prop =
5. m∠1 = m∠3 5. Subtr. prop.=
6. ∠1 ≅ ∠3 6. Def. ≅ ∠s

11. 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 Angles
If 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.

12. Example Find values of x and y that make the red
lines parallel and the blue lines parallel.
(x−40)° (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

13. Example Find values of x and y that make the red
lines parallel and the blue lines parallel.
(x−40)° (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

14. 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.

15. 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