Geometry Chapter 2: Reasoning and Introduction to Proof We can do this dude!

Inductive reasoning and conjectures Inductive reasoning is educated guessing based on experience. A conjecture is your educated guess. If your conjecture is false you need a counterexample to prove it is false. 2-1

Postulate/Theorem comparison Postulates are given rules or truths that are accepted without proving them. For example, You are making up a game to play with your friends. You decide that everyone takes a turn in sequence and the first person who reaches or passes 100 points is the winner. This is a ground rule you made up. Theorems are new rules created based on the original postulates. Theorems need to be proven in order to be true! Based on your game rules you can positively conclude that in any given game, there will be only one winner…no ties. What is the definition of a postulate? What is the definition of a theorem? 2-2

If you are blue then you are a smurf. If you are blue Then you are a smurf Conditional Statement is the if - then statement Hypothesis is the if statement Conclusion is the then statement

Lets try one:  If a triangle has three equal sides, then it is an equilateral triangle.  The conclusion is:  Then it is an equilateral triangle  The Hypothesis is:  If a triangle has three sides

Converse switch the if and the then statement. Conditional statement: If you got an A, then the teacher will be happy. If you have a four sided figure then you have a quadrilateral. Converse:  If the teacher is happy then you got an A.  If you have a quadrilateral, then you have a four sided figure.

Inverse: negate both the if and the then statement Conditional statement If it is raining then it is Friday If you are not happy, then it is Monday. Inverse If it is not raining, then it is not Friday. If you are happy, then it is not Monday.

Contrapositive: flip the if and the then and make opposite  Conditional statement  If it is a straight angle, then it is 180 degrees  If it is not supplementary, then it is not a 180 degrees  Contrapositive:  If it is not 180 degrees, then it is not a straight angle  If it is 180 degrees, then it is supplementary

Class work: write the converse, inverse, contrapositive, hypothesis and the conclusion 1.If you live in Dallas, then you live in Texas. 2.If you exercise regularly, then you are in good shape. 3.If the sum of two angles are 90 degrees, then the two angles are complementary. 4.If it is an acute angle, then the angle is less then 90 degrees. 5.If you are not a 100% satisfied, then return the product for a full refund.

Answer to number one: if you live in Dallas, then you live Texas 1.hypothesis: if you live in Dallas conclusion: then you live in Texas converse: If you live in Texas, then you live in Dallas. inverse: if you do not live in Dallas, then you do not live in Texas. contrapositive: if you do not live in Texas, then you do not live in Dallas.

Answer classwork 2. If you exercise regularly then you are in good shape. hypothesis: if you exercise regularly conclusion: then you are in good shape converse: if you are in good shape, then you exercise regularly. inverse: if you do not exercise regularly, then you are not in good shape. contrapositive: if you are not in good shape, then you do not exercise regularly.

3.If the sum of the two angles are 90 degrees, then the two angles are complementary. Hypothesis: if the sum of the two angles are 90 degrees. Conclusion: then the two angles are complementary. converse: if the two angles are complementary, then the two angles are 90 degrees. inverse: if the two angles are not 90 degrees, then the two angles are not complementary. contrapositive: if the two angles are not complementary, then the two angles are not 90 degrees

4.If it is an acute angle, then the angle is less then 90 degrees. hypothesis: if it is an acute angle conclusion: then the angle is less then 90 degrees converse: if the angle is less then 90 degrees, then it is an acute angle. Inverse: if it is not an acute angle, then the angle is more then 90 degrees. Contrapositive: if the angle is more then 90 degrees, then it is not an acute angle.

5. If you are not a 100% satisfied, then return the product for a full refund. Hypothesis: if you are not a 100% satisfied Conclusion: then return the product for a full refund Converse: if you return the product for a full refund, then you are not a 100% satisfied. Inverse: if you are satisfied, then you do not return the product for a full refund. Contrapositive: if you are not returning the product for a full refund, then you are satisfied.

Conditional Statement: p q Can be written in if then form. The if part is called the hypothesis or given The then part is called the conclusion or prove So hypothesis conclusion or given prove The converse statement is the reverse of the conditional statement: q p. The inverse statement is the negation of both parts of the conditional statement: ~ p ~ q. The contrapositive combines the converse and the inverse of the conditional statement: ~ q ~ p. 2-2

Deductive Reasoning Rules*: Law of detachment: If p q is true, and p is true, then q is true. Law of syllogism: If p q and q r are true, then p r is true. 2-3 *Deductive reasoning is making a conclusion based on facts previously known to be true.

Algebra Properties of equality : 1.a = a reflexive property of equality 2.If a = b then b = a symmetric property of equality 3.If a = b, and b = c, then a = c transitive property of equality 4.If a = b then a + c = b + c addition property of equality 5.If a = b then a - c = b - c subtraction property of equality 6.If a = b then ac = bc multiplication property of equality 7.If a = b then a/c = b/c division property of equality 8.If a = b and a = 4 then 4 = b substitution property of equality 9.a(b + c) = ab + ac distributive property of equality 2-4

Here is another algebra problem. This may seem difficult because you have to justify each move you make: Given: 12t = 4t – 24 Prove: t = -3 Proof: 12t = 4t – 24given 8t = -24subtract prop t = -3 division prop 2-4