OTCQ Match each picture to the words below. line segment ____ a. line ____ b. parallel lines ____ c....

OTCQMatch each picture to the words below.• line segment ____ a. • line ____ b. • parallel lines ____ c.

• intersecting lines ____ d. • ray ____ e. • point ____ f.

AIMS 3-4, 3-5, 3-6, 3-7

How do we define direct proofs and indirect proofs?

How do we start collecting our rule/postulates/theorems/properties

for proofs in geometry?

GPS 1, GPS 2, GPS 4,GRP 1 and GRP 7

OBJECTIVES1. SWBAT define direct and indirect proofs.2.SWBAT to define some rules/postulates/theorems/properties for our proofs in geometry.

Definitions

Direct Proof:An argument that starts with “given”

information and uses logic to arrive at a conclusion.

Indirect Proof:A proof that starts with a negation of the

statement to be proved and uses a counterexample or contradiction to arrive at a conclusion.

Direct Proof:An argument that

starts with “given” information and uses logic to arrive at a conclusion.

Indirect Proof:A proof that starts with

a negation of the statement to be proved and uses a counterexample or contradiction to arrive at a conclusion.

Direct Proof example:Sam is smart.All smart people drink water.Conclusion: Sam drinks

water.Indirect Proof:Space travel is not

impossible.People landed on the moon

in 1969.Conclusion: Space travel

must be possible.

Direct Proof example:

Sam is smart.All smart people drink water.

Conclusion: Sam drinks water.

Set of all water drinkers

Area shaded in blue representsall smart people.The smart people are asubset of the water drinkers.

Indirect Proof:Space travel is not possible.People landed on the moon

in 1969.Conclusion: Space travel

must be possible.

Indirect Proof.If this is the set ofspace travelers, then if space travel is impossible, it must be empty.Find anyone inside and space travel must be possible.

OBJECTIVES CHECK UP1. SWBAT define direct and indirect proofs.2.SWBAT to define some rules/postulates/theorems/properties for our proofs in geometry.

Recall Properties of Equality

1) Reflexive: a = a

2) Symmetric: If a = b then b = a.

3) Transitive:

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

4) Substitution:

If a = b, then a can be replaced by b.

Commutative PropertyCommutative Property of Addition: a + b = b + a

Commutative Property of Multiplication: ab = ba

Examples

2 + 3 = 5 = 3 + 2

3• 4 = 12 = 4 • 3

The commutative property does not work for subtraction or division!!!!!!!!

Associative PropertyAssociative property of Addition:

(a + b) + c = a + (b + c)

Associative Property of Multiplication:

(ab) c = a (bc)

Examples

(1 + 2) + 3 = 1 + (2 + 3)

(2 • 3) • 4 = 2 • (3 • 4)

The associative property does not work for subtraction or division!!!!!

Identity Properties

1) Additive Identity

a + 0 = a

2) Multiplicative Identity

a • 1 = a

Inverse Properties

1) Additive Inverse (Opposite)

a + (-a) = 0

2) Multiplicative Inverse (Reciprocal)

a 1

a 1

Multiplicative Property of Zero

a • 0 = 0

(If you multiply by 0, the answer is 0.)

The Distributive PropertyAny factor outside of expression enclosed within

grouping symbols, must be multiplied by each term inside the grouping symbols.

Outside left or Outside right

a(b + c) = ab + ac (b + c)a = ba + ca

a(b - c) = ab – ac (b - c)a = ba - ca

The Partition Postulate:

When 3 points A, B and C lie on the same line (are collinear) , we write ABC.

This implies:1. B is on line segment AC2. B is between A and C.3. AB + BC = AC

The Addition Postulate:

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

The Subtraction Postulate:

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