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Daniel Escobar

Daniel Escobar. Geometry Journal 2. Conditional statement. A conditional statement is a statement in the if-then form. P-Q

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Daniel Escobar

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  1. Daniel Escobar Geometry Journal 2

  2. Conditional statement • A conditional statement is a statement in the if-then form. P-Q • Examples: Ex.1: If I study, then I will pass my math class. Ex.2: If I arrive late to school, then I get a tardy. Ex.3: If I play videogames all night, then I forgot to do my homework. • Converse: A converse is the inverse of a conditional statement. ∼P∼Q. • Examples: Ex.1: If I passed my math class, then I studied. Ex.2: If I get a tardy, then I arrived late to school. Ex.3:If I forgot to do my homework, then I played videogames all night. • Inverse: Inverse are the same as conditional but hyp. + conclusion are not. ∼Q-∼P • Example: Ex.1: If I don’t study, then I don’t pass my math class. Ex.2: If I don’t arrive late to school, then I don’t get a tardy. Ex.3: If I don’t play videogames all night, then I don’t forget to do my homework. Contra-positive: Negates a converse. Examples: Ex.1: If I didn’t pass my math class, then I didn’t study. Ex.2: If I didn’t get a tardy, then I didn’t arrive late to school. Ex.3: If I didn’t forget to do my homework, then I didn’t play videogames all night.

  3. What is a counter-example? • A counter-example disproves a hypothesis. • Ex.1: “If I play guitar, then I am a musician.” (FALSE) • Counter-Example: I could play piano, and be a musician too. • Ex.2: √¯¯ (FALSE) Counter-Example: √¯¯ • Ex.3: “If I like AC/DC, then I love rock music” (FALSE) Counter-Example: I could be a KISS fan or a Rush fan, and hate AC/DC and still love rock music. 9 = 3 9 = -3

  4. What is a definition, perpendicular lines and lines perpendicular to plane? • Definition is a statement that describes a mathematical object and can be written as a true biconditional statement. • Perpendicular Lines are lines that intersect at 90∘angles • A line that intersects a plane at 90∘angles.

  5. What is a biconditional statement? • A biconditional statement is a statement that can be written in the form “p if and if q”. They are used to write definitions. They are important because they state that a certain hypothesis (“p”) can only have one conclusion (“q”). And they help to define. • Ex.1: A shape is a triangle if and only if, it is a three sided polygon. • Ex.2: Two angles are congruent if and only if their measures are equal.\ • Ex.: Two lines are perpendicular if and only if, when they intersect they form a 90∘angle.

  6. What is Deductive reasoning? • Deductive reasoning is the process of using LOGIC to draw conclusions from given FACTS, DEFINITIONS, and PROPERTIES. • Steps: 1.) Collect Data 2.) Look at Facts 3.) use logic- make conclusion • Deductive Reasoning has 2 Laws: • #1. Law of Detachment: If “p- q” is a true statement, Then if “p” is true, Then “q” must also be true. • 2#. Law of Syllogism: If P- Q and Q-R are both true statements then if “p” is true then “R” is true

  7. Laws of Logic There are two laws of logic: Law of Detachment, and Law of Syllogism. Law of Detachment: The Law of The Detachment states that If P-Q is a true statement, Then if P is true, Then Q must also be true. Ex.1: “If a shape is 3-sides, then it is a triangle” given: The shape is a triangle (therefore the shape is 3-sided” Ex.2: “If a student has an average of 85%, then he can sign up for the football team” Emilio can sign up for the football team (therefore he has a average of 85%) Ex.3: “If you/or another person is sick, you go to the doctor and ask for a prescription” Manuel went to the doctor and asked for a prescription. (therefore he was sick). Law of Syllogism: Law of Syllogism states that if p-q and q-r are true statements, then p-r is a true statement. Ex.1: “Studying gets good grades (p), good grades get you into good colleges (q), good colleges get you good jobs (r)” If you study and get good grades you will have a good job. Ex.2: “ if you are a murderer you go to jail (p), if you go to jail for murdering you are a bad person (q), if you are bad person you don’t go to heaven (r).” If you are a murderer you don’t go to heaven Ex.3: “Good guitarist practice long hours (p), with a lot of practice a guitarist can play difficult songs (q), professional guitarists play difficult songs.” If good guitarist practice long hours, they can become professional guitarists..

  8. Algebraic Proofs • This is the process when you do proofs by solving algebraic expressions. This is mostly used for proving algebraic expression. It is proven by algebraic properties. Properties like the subtraction, addition, division, etc. reason statement statement reason statement Reason Given Sbtraction Simplify Addition Smplify Division Simplify QED Given Addition property Simplify Subtraction prop. Simplify QED 5x-4=2x+8 -2x -2x 3x-4=8 +4 +4 3x/3 = 12 X=4 102=9f+90 -90 -90 12/9=9f/9 F=4/3 Given Subtraction Simplify Division Simplify QED 3x-6=2x+4 +6 +6 3x=2x+10 -2x -2x X=10

  9. Angle and Segment Properties of Equality and Congruence Properties of Equality Addition Property o.e: If A=b then a+c=b+c Subtraction Property o.e: If a=b then a-c=b-c Multiplication Prop o.e: If a=b then ac=bc Division Prop: If a=b and c≠0 then a/c =b/c Reflexive Prop: a=a Ex: 4=4 Symmetric Prop: if a =b then b=a Transitive Prop: If a=b and b=c then a=c Substitution Prop: If a=b then b can be substituted for “ a” Properties Of Congruence: Reflexive Prop: figure A≅figure A Ex: DE≅DE Symmetric Prop o.c: If figure A ≅ figure B, the figure B ≅ figure Ex if ∠A≅∠B then ∠B ≅∠ A Transitive Prop: If figure A≅figure B and Figure B≅ figure C then Figure A≅figure C

  10. Two-column proof • A two-column proof is used for organizing segments to have a better and more organized proof. You should start by labeling the right side with “reasons” and the left with “statements” Statements Reasons ex1 reasons statement statements reason 3x-8=19 +8 +8 3x=27 3 3 X=9 Given Addition simplify Subtraction Simply Division 4x-2=6x+8 +2 +2 4x=6x+10 -6x -6x -2x=10 X=-5 3x-6=12 -6 -6 3x=6 3 3 X=2 Given Subtraction Simplify Addition Simplify QED Given Addition Simplify Subtraction Simplify Division simplify

  11. Linear Pair Postulate • What is Linear pair Postulate? • A Linear Pair Postulate (LPP), is All linear postulates are supplementary angles • Ex1 2 Ex2 • Ex3 1 BA and BC form a line ∠1 and ∠2 are a pair of adjacent angles with opposite rays

  12. congruent complements and supplements theorems • Congruent Complements Theorem: This theorem states that If 2 angles are complementary t the same angle (or two congruent angles), then the two angles are congruent. • Ex1 ∠1=and ∠2 add up to∠2=90∘ • Ex2 ∠1 and ∠2 are complementary • Congruent Supplements Theorem: This theorem states that If two angles are supplementary to the same angle (or to two congruent angles) then the two angles are congruent. • Ex1: ∠1 +∠2=180∘ • Ex2: ∠1 and ∠2 are supplementary 2 1 1 2

  13. Vertical Angle Theorem • The Vertical Angle Theorem states that Vertical angles are congruent. • Ex: ∠A ≅∠B ex:∠1≅∠2,∠3≅∠4 • Ex ∠A≅∠B Ex: C D A B A ∠C≅∠D D C B

  14. Common Segment Theorem • This theorem states that if given collinear points A,B,C, D Ex: AB≅CD then AC≅BD • Ex EF≅GH then EG≅FH Ex IJ≅KL then IK≅JL A B C D E F G H I J K L

  15. THANK YOU FOR YOUR PATCIENCE

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