OTCQ Match 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

OBJECTIVES 1. 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 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 represents all smart people. The smart people are a subset of the water drinkers.

Indirect Proof: Space travel is not possible. People landed on the moon in Conclusion: Space travel must be possible. Indirect Proof. If this is the set of space travelers, then if space travel is impossible, it must be empty. Find anyone inside and space travel must be possible.

OBJECTIVES CHECK UP 1. 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 Property Commutative Property of Addition: a + b = b + a Commutative Property of Multiplication: ab = ba Examples = 5 = = 12 = 4 3 The commutative property does not work for subtraction or division!!!!!!!!

Associative Property Associative 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)

Multiplicative Property of Zero a 0 = 0 (If you multiply by 0, the answer is 0.)

The Distributive Property Any factor outside of expression enclosed within grouping symbols, must be multiplied by each term inside the grouping symbols. Outside leftorOutside 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 AC 2.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.