Transcript Geometry chap. 2

• 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.
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.
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°.
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
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.
2. 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.
3. 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.
Law of detachment: if conclusion then 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.
If p then q and q then r, they are true statements , then p then r is a true statement..
Examples: determine if each conjecture is true by the law syllogism.
1. Given: if a number is divisibal by 4 then it is divisible by 2.
Conjecture: if the number od divisible by 4 then it is even.
2. Given: If a number is divisible by 12 it is divisble by 2,3,4 and 6.
Conjecture: all the numbers divisible by 12 are odd and even.
3. Given: If a win all of my classes i go to dubai. I won all of my classes.
Conjecture: i am going to dubai
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.
Solve these equations and justify each step:
-11=3n+1
-1
-1
property
-12=3n
-12/3
ofequality
-4=n
N= -4
given equation
subtraction property of equality
450=3.6p given equation
450/3.6=3.6p/3.6 division
simplify
division property of equality
125=p
simplify
p=125p symmetric property
-13=2n+1
-1
-1
-14=2n
-14/2
-7=n
N=-7
given equation
subtraction property of equality
simplify
division property of equality
simplify
symmetric property of equality
simplify
symmetric property of equality
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.
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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 is congruent to figure B, then figure b is congruent to figure
a.
If <1 is congruent to <2 then <2 is congruent to <1
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XY is congruent to VW so VW is congruent to XY.
You can do a 2 column proof by divading a square into to parts the left part will be
were you list the steps of each problem and in the right you match the reason for each
step.
Examples:
Statements :
1. <1 and <2 form a linear pair.
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2. BA and BC form a line.
3. m<ABC=180
4. a.________
5.b._________
6. <1 and <2 are suplementary.
reasons:
1. given
2. def. of a lin. pair
3. def if a straight <
4. < add. Post.
5. subst. steps 3,4
Given <1 and <2 are suplementary and <2 and < are supplementary
Prove <1 is congruent to <3
proof
reason
statements
1. a._______
2. m<1+m<2=180
m<2+m<3=180
3. b._____
4. m<2=m<2
5. m<1=m<3
6. d._______
1.
2.
3.
4.
5.
6.
Given
def. of supp.
subst.
reflex. Prop. Of =
c._______
def. of congruent angles
Given :<1 and 2 are right angles.
Prove: <1 and <2 are congruent
statements
1.
2.
3.
4.
<1 and <2 are right angles.
m<1=90 m<2=90
m<1= m<2
<1 congruent to <2
reasons
1.
2.
3.
4.
Given
def. of right angle
trans, property of equality
def. of congruent angles
Linear pair postulate: all linear pairs of angles are supplementary.
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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.
All vertical angles that aren't adjacent are congruent.