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Geometry Journal 2

Geometry Journal 2. Andres Cofiño. Conditional if-then statement. Conditional if-then statement is a statement that has to parts: a hypothesis and a conclusion. The conclusion always states the hypothesis. If the hypothesis happens, then the conclusion hap p ens .

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Geometry Journal 2

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  1. Geometry Journal 2 Andres Cofiño

  2. Conditional if-then statement • Conditional if-then statement is a statement that has to parts: a hypothesis and a conclusion. The conclusion always states the hypothesis. If the hypothesis happens, then the conclusion happens. Converse: when you switch the hypothesis and the conclusion of a conditional statement. Inverse: same as conditional but hypothesis and conclusion have a no/not. Ex. If an ice-cream is under the sun for to long, then it will melt. If you use your computer for a long time, then the battery will run out. It you cheat on the game, then you’ll be disqualified.

  3. Counterexample • Is an example to disprove something or to show something is not true. The opposite of a number is always positive. Counterexample: The opposite of 9 is - 9, a negative number. Students that are in 12 grade are the smartest in all school. Counterexample: There can be an 11th grader smarter than a kid that is failing classes in 12 grade. Every line is straight and never end. Counterexample: Curved line or a segment

  4. Definition of Definition • The statement or significance of phrase or word. Ex. The definition of a point is: a mark in space represented with a dot. The definition of a line is: a one dimensional set of points aligned that can go forever in both directions with no width. The definition of a plane is: a Two dimensional figure withlength and widthdefinedbythreepoints.

  5. Bi-Conditional Statement A bi-conditional statement is a statement that its conditional statement and its converse are true statements. They are true even if changing the side of the conclusion and the hypothesis. Bi-conditional statements are used with the phrase/word iff or if and only if. They are important because they can give us an accurate definition of something. Ex. Two angles are congruent iff they have equal measures. It is a right angle iff its measure is 90 degrees. It is an acute angle iff its measure is below 90 degrees and above 0 degrees.

  6. Deductive Reasoning Deductive reasoning is the process of using logic to draw conclusions from facts and definitions. It is used basically by looking at the facts of certain statement to find the conclusion you want to end up with. Symbolic notation is the use of symbols instead of words in simple expressions. Ex. Equal: = Minus: - Plus: +

  7. Deductive Reasoning examples: 1. All lemons are fruits All fruits grow on trees So based on that, all lemons grow on trees. 2. All 12 graders are single Carlos is single, Therefore, Carlos is a 12 grader. 3. The members of the Fuentes family are Jose, Juana and Oscar. Jose is fat Juana is fat Oscar is fat Based on this, all members of the Fuentes family are fat.

  8. Laws of Logic • Law of Detachment: if P then Q is a true statement, then if P is true, the Q must also be true. Ex. 1.If an angle is obtuse, then it cannot be acute. Angle A is obtuse. Therefore, Angle A cannot be acute. 2. If is is raining, Then you will get wet. It is raining. So then you will get wet. 3. If it is Wednesday, we don’t have class. It is Wednesday. Then, We don’t have class.

  9. Law of Syllogism: If P then Q and Q then R are both true statements, then if P is true then R is true. Ex. 1. If the electric power is cut, then the microwave doesn't work. If the microwave doesn't’ work, then we cannot warm our food. So if the electric power is cut, then we cannot warm our food. 2. If the canal is open, then the ship can go through the canal. If the ship can go through the canal, then the company can transport their goods. Then if the canal is open, then the company can transport their goods. 3. If a triangle has angles of 30 degrees and 60 degrees, then its third angle is 90 degrees. If an angle in a triangle is 90 degrees, then it is a right triangle. So then if a triangle has angles of 30° and 60°, then it is a right triangle.

  10. Algebraic Proofs Is a proof that uses algebraic properties to solve a problem step by step to validate your answer. To do an algebraic proof you can make a two-column proof, paragraph proof, flowchart proof by explaining each step you do to solve a problem.

  11. Algebraic Proofs Examples: reason statement 3x-6=2x+4 3x-8=19 Given Given +8 +6 +8 +6 Addition Property Addition Property 3x=2x+10 3x=27 Simplification Division Property -2 -2 /3 /3 Simplification X=9 X=10 simplification statement reason Subtraction Property

  12. statement reason 5x-4=2x+8 Given +4 +4 Addition Property 5x=2x+12 Simplification -2 -2 3x=12 Simplification /3 /3 Division Property X=4 simplification Subtraction Property

  13. Segment and Angle Properties of Equality and Congruence Segment: Equal- CV = TR, and TR = XY, then CV = XY Congruent- CV is congruent to TR and TR congruent to XY then CV is congruent to XY Angle: Equal- m∠V = m∠X, then m∠X = m∠V Congruent- If ∠T congruent ∠S, then ∠S congruent ∠T

  14. Examples of segment and angle properties If angle P is congruent to angle Q and angle Q is congruent to angle R, then angle P is congruent to angle R. np=npnp is conguent to np M<2=m<2 <s is congruent to <s

  15. Two-Column Proofs To do a two-column proof, you need to right your statements on the left and your reasoning's on the right. Statements are the steps you make to solve a problem and reasons are what defines them. statement reason C=9f+90 Given 102=9f+90 Substitution Property -90 -90 Subtraction Property Division Property /9 /9 C=102 12=9f simplification F=1.33333… Simplification

  16. statement statement reason reason Given Given M<1=90 degrees and m<2=90 degrees <1 and <2 are supplementary LPP Def. right angle <1 and <2 are rightangles M<1=m<2 Transitive Property Congruent angles supplementary <1 and <2 are congruent <1 and <2 are right <1 is congruent to <2 Def. of Congruence Right angles

  17. LPP Linear Pair postulate states that all linear pairs of angles are supplementary. 5 175 90 90 50 130

  18. Congruent Supplements and Complements Theorem Congruent Supplements Thm: If two angles are supplementary to the same angle or to congruent angles, then they are congruent. Congruent Complements Thm: If two angles are complementary to the same angle or to congruent angles, then they are congruent.

  19. Vertical Angles Vertical Angles theorem states that all vertical angles are congruent. 100 1 80 90 2 80 90 100 4 90 90 3 <1 and <3 are congruent <2 and <4 are congruent

  20. Common Segments Theorem This theorem states that 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. Ex. 1. If Chicago to Detroit is the same as Seattle to Los Angeles then Chicago to Seattle is the same as Detroit to Los Angeles. 2. The distance from Ihopto Starbucks is the same as from McDonalds to Burger King. Then the distance from Ihopto McDonalds is the same as from Starbucks to Burger King. 3. The distance from Mario’s playground to Juan’s is the same as the distance from Jose’s playground to Tom’s playground. Then from Mario’s playground to Jose’s playground is the same distance as from Juan’s playground to Tom’s playground.

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