Geometry chapter 2 journal

1. Geometry chapter 2 journal By: Alejandro Hernández

2. Conditionalstatement: • Every conditional statement has a conclusion and a hypothesis. • Example of a hypothesis and a conclusion: • If a dog is yellow and has a big body, then it is a golden retriever. • Hypothesis: If a dog is yellow and has a big body. • Conclusion: then it is a golden retriever. • converse: is the statement that goes from from q to p, q stands for conclusion and p for hypothesis. • inverse: is the statement that is making the hypothesis and conclusion false. • contrapositive : is the statement that negating and exchanging the hypothesis and conclusion. • Examples: • Converse: if a mammal has four legs, then it is a dog. • Inverse: if a mammal is not a dog, then it does not have four legs. • Contrapositive: if a mammal does not have four legs, then it is not a dog.

3. Counterexample Counterexample: is a example that proves that a conjecture is false. Examples: The smallest country in central America is costa Rica. The biggest country in central America is El Salvador. Belize is part of Guatemala.

4. Definition A definition is when you say or define mathematical object and can be written as a true biconditional statement. Examples: Geometry: The mathematics of the properties, measurement, and relationships of points, lines, angles, surfaces, and solids. Algebra: A branch of mathematics that substitutes letters for numbers.  Right angle: A right angle is an angle which is equal to 90°.

5. Biconditionalstatement Is the statement that it is written in the form of, if and only if. Examples: 1. An angle is acute when its measure is greater than o and less than 90 degrease. Biconditional: An angle is acute if and only if its measure is greater than 0 and less than 90 degrease. 2. Usnavi is a member when he has paid the Q5. Biconditional: usnavi is a member if and only if he has paid the Q5. 3. 5x+5= 12, then x= 5 Byconditional: 5x+5=12 if and only if x=5

6. Deductivereasoning Definition: Is when you use logic to draw a conclusion of facts given, definitions and properties. Symbolic notation: is when you are writing a symbol instead of a word. Examples: 1. per every class of the 9th grade their are at least 15 blackberries so using logic we assume that in total their are more than 75 blackberries in the whole 9th grade. Per every grade level of high school their are at least 20 people that ride the bus to their homes so using logic we say that more than 80 persons in high school use the bus to get to their homes. per every class of the 9th grade their are at least 2 blond people so using logic we say their are more than 10 people with blond hair in the 9th grade.

7. Laws of logic Law of detachment: if conclusionthen hypothesis then it is a true statement and if the hypothesis is true then the conclusion is true. Examples of laws of detachment: Determine if the conjecture is valid by the law of detachment: 1. Given: The rules of high school say that if you have a lower grade than 65 on any class you may not play on any sport team, so Alejandro passed all of his classes with a greater grade of 70. Conjecture: Alejandro is able to play on any sport team. 2. Given : They say that if an angle is equal to 90 degrease then it is a right angle. Conjecture: it is a right angle. 3. Given: if two segments have the same measure then they are congruent. Conjecture: they have the same measure so they are congruent.

8. Law of syllogism If p then q and q then r, they are true statements , then p then r is a true statement.. Examples: determine ifeachconjectureis true bythelawsyllogism. 1. Given: if a numberisdivisibalby 4 thenitis divisible by 2. Conjecture: ifthenumberod divisible by 4 thenitiseven. 2. Given: If a numberis divisible by 12 itisdivisbleby2,3,4 and 6. Conjecture: allthenumbers divisible by 12 are odd and even. 3. Given: If a winall of myclasses i gotodubai. I won all of myclasses. Conjecture: i am goingtodubai

9. Algebraicproofs: We all have solved a algebra equation and all of these equations use the distributive property for example. The first thing to do is solve the equation given, then prove each step with a property. Properties: Addition property of equality: if a= b, then a+c=b+c Subtraction property of equality: if a= b, then a-c=b-c Multiplication property of equality: if a=b, then ac= bc Division property of equality: if a=b and c=0, then a/c= b/c Reflexive property of equality: a=a Symmetric property of equality: if a= b and b= a Transitive property of equality: if a=b and b=c then a=c Substitution property of equality: if a=b, then b can be substituted for a in any expression.

10. Algebraicexamples: Solve these equations and justify each step: -11=3n+1 given equation 450=3.6p given equation -1 -1 subtraction property of equality 450/3.6=3.6p/3.6 division property -12=3n simplify 125=p simplify -12/3 division property of equality p=125p symmetric property ofequality -4=n simplify N= -4 symmetric property of equality -13=2n+1 given equation -1 -1 subtraction property of equality -14=2n simplify -14/2 division property of equality -7=n simplify N=-7 symmetric property of equality

11. Segmentangleproperties of equality and congruency Transitive:if a figure A is congruent to figure B if it is congruent to figure C, then figure a is congruent to figure c. __ __ __ __ __ ___ If pq is congruent to rs and rs is congruent to tu then pq is congruent to TU. <1 is congruent to <2 and <2 is congruent to <3 so <1 is congruent to <3 symetric: Congruency: if a figure A iscongruentto figure B, then figure b iscongruentto figure a . If <1 is congruent to <2 then <2 is congruent to <1 __ ____ ___ __ XY is congruent to VW so VW is congruent to XY.

12. 2 culumnproof You can do a 2 columnproofbydivading a squareintotopartstheleftpartwill be wereyoulistthesteps of eachproblemand in therightyou match thereasonforeachstep. Examples: Statements : reasons: <1 and <2 form a linear pair. 1. given __> __> 2. def. of a lin. pair 2. BA and BC form a line. 3. def if a straight < 3. m<ABC=180 4. < add. Post. 4. a.________ 5. subst. steps 3,4 5.b._________ 6. <1 and <2 are suplementary.

13. Given <1 and <2 are suplementary and <2 and < are supplementary Prove <1 is congruent to <3 proof reason statements a._______ m<1+m<2=180 m<2+m<3=180 3. b._____ 4. m<2=m<2 5. m<1=m<3 6. d._______ Given def. of supp. subst. reflex. Prop. Of = c._______ def. of congruent angles

14. Given :<1 and 2 are right angles. Prove: <1 and <2 are congruent statements reasons Given def. of right angle trans, property of equality def. of congruent angles <1 and <2 are right angles. m<1=90 m<2=90 m<1= m<2 <1 congruent to <2

15. LPP!!! Linear pair postulate: all linear pairs of angles are supplementary. http://content.tutorvista.com/maths/content/geometry/lines%20angles%20triangles/images/img61.gif

16. Congruent, complents and Supplements theorems Congruent Complements theorem: If two angles are complementary to the same angle then they are congruent. Congruent Supplements theorem: If two angles are supplementary to the same angle then they are congruent.

17. Vertical angles theorem All vertical angles that aren't adjacent are congruent.