Conjecture: an educated guess Inductive reasoning: Looking at several specific situations to arrive at a conjecture. Juan, Pedro, Teres and Ines are ISP students and they are 10, 17, 15 and 16 years old. Conjecture: All ISP students are 18 years old or less.

Counterexample A conjecture may be true or false It takes only one false example to show a conjecture is not true. The false example is called a counterexample. Orlando is an ISP student, and he is 19 years old. Then, not all ISP students are 18 years old or less.

Conditional Statement Conditional statement (if-then statement) : hypothesis + conclusion (p  q) Example: If we do well on the test we are going to improve our grade in the subject.

Converse The converse of a conditional is written by interchanging the hypothesis and the conclusion. (q  p) Example: If we improve our grade in this subject, then we are going to do well on the test

Inverse Negation: the denial of a statement. (p) Inverse can be formed by negating both, the hypothesis and the conclusion. (p q) Example: If we don’t do well on the test, then we are not going to improve our grade in this subject.

Contrapositive Contrapositive: Negating the hypothesis and the conclusion of the converse of the given conditional. (no q  no p) Example: If we don’t improve our grade in this subject, then we are not going to do well on the test.

Postulates Through any two points there is exactly one line Through any three points not on the same line there is exactly one plane A line contains at least two points A plane contains at least three points not on the same line If two points lie in a plane, then the entire line containing those two points lies in that plane If two planes intersect, then their intersection is a line.

Deductive and Inductive Reasoning Inductive reasoning uses examples to make a conjecture or rule. From specific to general. Deductive reasoning uses a rule to make a conclusion. From general to specific. All ISP students are 18 years old or less. Juan is an ISP student, so he is 18 years old or less.

Law of detachment Law of detachment: If p  q is a true conditional and p is true, the q is true. If two lines are parallel, they don´t intersect each other. (True statement) l If l and m are parallel, they don´t intersect each other. (Application of the law of detachment) m

Law of Syllogism Law of syllogism: If p  q and q  r are true conditionals, the p  r is also true. If two lines are perpendicular they form right angles. (True statement) If two lines form right angles, they divide the plane in four equal angles. l m If two lines are perpendicular they divide the plane in four right angles. (Application of the law of Syllogism)

Some properties from algebra applied to geometry Property Segments Angles Reflexive Symmetric Transitive m<1 = m<1 PQ=QP If AB= CD, then CD = AB. If m<A = m<B, then m<B = m<A If GH = JK and JK = LM, then GH = LM If m<1 = m<2 and m<2 = m<3, then m<1 = m<3

Examples Name the property of equality that justifies each statement. Reasons If AB + BC=DE + BC, then AB = DE m<ABC= m<ABC If XY = PQ and XY = RS, then PQ = RS If (1/3)x = 5, then x = 15 If 2x = 9, then x = 9/2 Subtraction property (=) Reflexive property (=) Substitution property (=) Multiplication property (=) Division property (=)

Example 2 Justify each step in solving 3x + 5 = 7 2 Statement Reasons The previous example is a proof of the conditional: If 3x + 5 = 7, 2 then x=3 3x + 5 = 7 2 Given 2(3x + 5) = (7)2 2 Multiplication property (=) 3x + 5 = 14 Distributive property (=) Subtraction property (=) 3x = 9 x = 3 Division property (=) This type of proof is called a TWO-COLUMN PROOF

Verifying Segment Relationships Five essential parts of a good proof: State the theorem to be proved. List the given information. If possible, draw a diagram to illustrate the given information. State what is to be proved. Develop a system of deductive reasoning. (Use definitions, properties, postulates, undefined terms, or other theorems previously proved).

Theorem 2.1 Congruence of segments is reflexive, symmetric, and transitive. Reflexive property: AB  AB. Symmetric property: If AB  CD, then CD  AB Transitive property: If AB  CD, and CD  EF, then AB  EF Abbreviation: reflexive prop. of  segments symmetric prop. of  segments transitive prop. of  segments

Verifying Angle Relationships Theorem 2-2 Supplement Theorem: If two angles form a linear pair, then they are supplementary angles Theorem 2-3: Congruence of angles is reflexive, symmetric and transitive. Abbreviation: reflexive prop. of  <s symmetric prop. of  <s transitive prop. of  <s

Theorem 2-4 and 2-5: Theorem 2-4 Angles supplementary to the same angle or to congruent angles are congruent: Abbreviation: <s supp. to same < or  <s are  Theorem 2-5: Angles complementary to the same angle or to congruent angles are congruent. Abbreviation: <s comp. to same < or  <s are 

Abbreviation: All rt. <s are  Theorem 2-6, 2-7 and 2-8 Theorem 2-6 : All right angles are congruent Abbreviation: All rt. <s are  Theorem 2-7: Vertical angles are congruent. Abbreviation: Vert. <s are  Theorem 2-8: Perpendicular lines intersect to form four right angles. Abbreviation:  lines form 4 rt. <s