Maricarmen Hernandez Geo Journal #2

Conditional If-Then Statements: A conditional if-then statement is when a sentence has a hypothesis (if) and a conclusion (then). If is shown by P, and then is shown by Q. If I eat my vegetables, then I´ll be strong. (P)  (Q)

Conditional If-Then Statements example: If I eat my vegetables, then I´ll be strong. (P)  (Q) Converse- When we switch the hypothesis and the conclusion of a conditional statement. Q  P If I´m strong, then I ate my vegetables. Inverse- Same as conditional but hypothesis & conclusion are NOT. If I didn’t eat my vegetables, then I´m not strong. Contrapositive- Negative the converse. If I´m not strong, then I didn’t eat my vegetables.

Counter-examples: When an example is used to disprove a hypothesis. Mammals don’t lay eggs. The platypus does. All water animals aren´t mammals. Dolphins are mammals. When you add, the result is always positive = -4

Definition: A definition is a biconditional statement that defines what something is. A number is prime if it can only be divided by one and itself. It’s a statement if the line has a start and an end. It’s a triangle if it has 3 sides.

Bi-conditional Statements: Its when both the conditional and the converse statements are true. If and only if (iff). They are used to show when both the conclusion and the hypothesis depend of each other and both are true. They are important because we use it in an every day basis, either consciously or unconsciously. We use it for logic and in proofs. Examples: o A shape is a triangle iff it has 3 sides. o A line iff it goes on forever on both directions. o A ipod iff its designed by Apple and produces music.

Deductive Reasoning: A way of reasoning in which you use data and logic to make conclusions. Collect Data Look at the Facts Use logic – make the conclusion. Ex)Research shows that 5 of 10 kids in the class have blue pens, so we can say 50 out 100 kids in the grade have blue pens. Research shows that 1 out of every 2 kids in our grade have cell phones, so we can say that half the grade have cell phones. Research says that one of every 3 kids in CAG have crocs, so we can say that in a class of 21, 7 kids own crocs. Symbolic Notation: When a symbol is used instead of the word/words. Ex) times: x, equal: =

Laws of logic: Law of detachment Law of syllogism Law of Detachment: If P  Q and P is true, then Q must be too. Ex1) You must be at least a meter and a half tall to ride the rollercoaster. Sandra is 1.20 meters tall. Sandra can´t ride the rollercoaster. Ex2) If you are over 10 years old, you must pay full price to enter the park. Sean is 13 years old. Sean will pay full price. Ex3) You need at least 65% in a class to pass the grade. Tawny´s grade is 50%. Tawny did not pass the grade.

Law of Syllogism: If P  Q and Q  R are true, then P  R is true. Ex1) If you live in Guatemala city, then you live in Guatemala. If you live in Guatemala, then you live in Central America. I live in Guatemala city, therefore I live in Central America. Ex2) If you have Geometry, then you are in high school. If you are in high school, you have more than 13 years old. Ron is in Algebra, so he has more than 13 years. Ex3) If you have a car, then you can drive. If you can drive, you have a license. Cedric has a car, so he has a license.

Algebraic Proofs & Properties: Addition Property If a=b, then a+b=c Subtraction Property If a=b, then a – c is b – c Multiplication Property If a=b, then ac=bc Division Property If a=b & c don’t equal zero, then a/c = b/c Reflexive Property a=a Symmetric Property If a=b then b=a Transitive Property if a=b and b=c, then a=c Substitution If a=b, then b can be replaced by a in any situation Step by step arguments that prove something.

StatementReason Ex1) 4x-2=6x+8 4x-2=6x x=6x+10 -6x -2x=10 -2 X=-5 Given Addition Simplify Subtraction Simplify Division Simplify Q.E.D

StatementReason Ex2) x+8=9 X+8= X=1 Given Subtraction Simplify Q.E.D

StatementReason Ex3) 3x-8=19 3x-8= x=27 3 x=7 Given Addition Simplify Division Simplify Q.E.D

Properties of congruency and equality: Transitive: If AB = CD, and CD = EF, then AB = EF Symmetric: If AB = CD, then CD = AB. Reflexive: : AB = AB, AB congruent to AB

2 Column Proof It’s a way to prove something in a chart, you put the statement in the left and the reason in the right. 1.Write down important information 2.Draw a picture 3. Mark what's given. 4.Write down the info and pictures. 5. GO FOR IT!!!

StatementReason Ex1) Given: LPP-if 2 angles for a linear pair then they are supplementary. <1 and <2 are linear pair. <1 and <2 form a line. M<1 and M<2=180 <1 and <2 are supplementary. Given LPP def. Def. straight angle Def. supplementary Q.E.D

StatementReason Ex2) Right angle theorem- all right angles are congruent. Give: <1 and <2 are right angles. <1 and <2 are right angles. M<1= 90 and M2=90 M<1 = M<2 <1 is congruent to <2 Given Def. right angle. Transitive Congruent def. Q.E.D

StatementReason Ex3) Given: <1 and <2 are supplementary. <1 and <2 are supplementary. Prove: <2 Is congruent to <3. if 2 angles are supplementary to the same angle, then they are congruent. <1 and <2 are supplementary. <1 and <3 are supplementary. M<1+M<2=180 M<1+M<3=180 M<1+M2=M<1+M<3 -M<1 M<2 is congruent to M<3 Given Def. Supplementary Transitive Subtraction Def. of Congruent Q.E.D

LPP: (Linear Pair Postulate) all linear pairs of angles are supplementary.

Vertical Angles Theorem: All non adjacent vertical angles are congruent.

Congruent Segments Theorem: if points A, B, C, and D are all collinear, then segment AB is congruent to segment CD then segment AC is congruent to segment BD. Ex1) If angle AB is 50°, then CD is 50°. AC is 30°, then BD is 30°. Ex2) If the distance between Spain and Portugal are the same as Germany to France, then it’s the same between Spain and Germany, and Portugal and France. Ex3) If the distance between Mexico and Guatemala is the same as El Salvador to Honduras, then it’s the same between Mexico and El Salvador as Guatemala and Honduras.

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