Download 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 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.
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
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 with length and width defined by three
points.
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.
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: +
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.
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.
• 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.
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 twocolumn proof, paragraph proof, flowchart proof
by explaining each step you do to solve a
problem.
Algebraic Proofs Examples:
statement
3x-8=19
+8 +8
3x=27
reason
Given
Addition Property
Division Property
/3 /3
X=9
statement
simplification
reason
3x-6=2x+4
+6 +6
Given
Addition Property
3x=2x+10
-2 -2
Simplification
X=10
Simplification
Subtraction Property
statement
5x-4=2x+8
+4 +4
5x=2x+12
-2 -2
reason
Given
Addition Property
Simplification
Subtraction Property
3x=12
/3 /3
X=4
Simplification
Division Property
simplification
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
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=np
np is conguent to np
M<2=m<2
<s is congruent to <s
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
C=9f+90 C=102
102=9f+90
-90
-90
12=9f
/9 /9
F= 1.33333…
reason
Given
Substitution Property
Subtraction Property
Simplification
Division Property
simplification
statement
<1 and <2 are right
M<1=90 degrees and m<2=90
degrees
M<1=m<2
<1 is congruent to <2
statement
<1 and <2 are congruent
<1 and <2 are supplementary
<1 and <2 are right angles
reason
Given
Def. right angle
Transitive Property
Def. of Congruence
reason
Given
LPP
Congruent angles supplementary
Right angles
LPP
Linear Pair postulate states that all linear pairs of
angles are supplementary.
175
90
90
50
130
5
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.
Vertical Angles
Vertical Angles theorem states that all vertical
angles are congruent.
1
100
80
2
80
100
3
90
90
4
90
90
<1 and <3 are congruent
<2 and <4 are congruent
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 Ihop to Starbucks is the same as from McDonalds to
Burger King. Then the distance from Ihop to 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.