geometry proofs math 416. time frame definition definition congruent triangles congruent triangles...
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Time FrameTime Frame
DefinitionDefinition Congruent TrianglesCongruent Triangles Axiom & ProofsAxiom & Proofs PropositionsPropositions
DefinitionsDefinitions
Geometric ProofsGeometric Proofs The essence of pure mathematicsThe essence of pure mathematics The creative and artistic center of The creative and artistic center of
mathmath The ability to explain in a detailed The ability to explain in a detailed
concise logical manner how a concise logical manner how a proposition (problem) is either true or proposition (problem) is either true or false. false.
Definitions (con’t)Definitions (con’t)
Detailed – hard factsDetailed – hard facts Concise – short to the pointConcise – short to the point Logical – set of rules based on reasonLogical – set of rules based on reason A proof generally falls back to things A proof generally falls back to things
that are either known, accepted or that are either known, accepted or already proven. This is how we attain already proven. This is how we attain knowledge knowledge
Gaining KnowledgeGaining Knowledge
PropositionPropositionPropositionProposition
Proposition
Definition
Axiom Thoerem
En
lighte
nm
ent
DefinitionsDefinitions
Definition: You define something Definition: You define something once you once you identifyidentify its essential its essential characteristicscharacteristics
For example, triangle – a two For example, triangle – a two dimensional polygon with three sidesdimensional polygon with three sides
Not Must
AxiomAxiom
Axioms: An obvious statement that is Axioms: An obvious statement that is acceptable without proofacceptable without proof
For example, the shortest distance For example, the shortest distance between two points is a straight linebetween two points is a straight line
PropositionsPropositions
Propositions are statements that Propositions are statements that require proof require proof
Once proven they are called Once proven they are called theorems theorems
For exampleFor example1
23
Proof
STATEMENT AUTHORITIES
<1 + <3 = 180°<2 + < 3 =
180°<1 = <2 = 180
DEFINITIONDEFINITION
ALGEBRA
Theorums Theorums
This proposition now becomes a theorem This proposition now becomes a theorem Hence, vertically opposite angle theorem Hence, vertically opposite angle theorem Theorems can be used in a proof as an Theorems can be used in a proof as an
authority authority Definitions must Definitions must
use terms that are already defineduse terms that are already definedBe reversible once you have the Be reversible once you have the
characteristics you have the objectcharacteristics you have the objectnot give unnecessary informationnot give unnecessary information
Examples #1 of DefinitionsExamples #1 of Definitions
are belingas
Which of the following is a belingas?
Definition: A belingas is a shape with a dot on a vertex
Example #2 of a DefinitionExample #2 of a Definition
Are GatusWhich of the following is a Gatu?
Definition: A Gatu is a shape with at least one curved side
Stencil #1
AxiomsAxioms
A statement not requiring proofA statement not requiring proof A whole is equal to the sum of its A whole is equal to the sum of its
partpart Completion Completion
ADB
C
< ABD = <ABD + <CBD• Any quantity can be replaced by another equal quantity
AxiomsAxioms
Replacement… Replacement… If a + b = cIf a + b = c AND b = q AND b = q Then a + q = Then a + q = The shortest distance between two points is The shortest distance between two points is
a straight linea straight line Only one line can pass through the same Only one line can pass through the same
two pointstwo points Given a point and a direction, only one line Given a point and a direction, only one line
with that direction can pass through the with that direction can pass through the pointpoint
c
Easiest thing to do is to assign numbers to letters… a=0;b=4;c=4;q=4
PostulatesPostulates Theorems we will not prove are called Theorems we will not prove are called
postulates specifically the congruence postulates specifically the congruence postulatespostulates
Hypothesis: Given two triangles with Hypothesis: Given two triangles with corresponding sides equal we say corresponding sides equal we say
CONC: Two triangles are congruent CONC: Two triangles are congruent
ABC YZX
By S S S
A
B C ZY
X
PostulatesPostulates
Hypothesis: Given two triangles with two Hypothesis: Given two triangles with two corresponding sides equal and the corresponding sides equal and the contained angle equalcontained angle equal
Conclusion: The two triangles are Conclusion: The two triangles are congruentcongruentA
Y
Z
X
CB
ABC ZXYBy SAS
°
°
PostulatesPostulates Hypothesis: Given two triangles Hypothesis: Given two triangles
with two corresponding angles with two corresponding angles equal and the contained side equal and the contained side equalequal
Conclusion: The two triangles are Conclusion: The two triangles are congruentcongruentA
ZY
X
CB
O
O X X
ABC ZXY
By ASA
Do #2
TheoremsTheorems
Once again we will not prove Once again we will not prove But you may be required toBut you may be required to You should be able to You should be able to
TheoremsTheorems
The 90° completion theorem or the The 90° completion theorem or the complementary angle theoremcomplementary angle theorem
The 180° Completion TheoremThe 180° Completion Theorem
xy
HYP: Diagram
CONC < X + <Y = 90°
x y
HYP Diagram
CONC <x + <y = 180
Vertically Opposite Angle Vertically Opposite Angle TheoremTheorem
1
432
Conclusion < 1 = < 2
< 3 = <4
Isosceles Triangle TheoremIsosceles Triangle Theorem
1
Conclusion<1 = <2
2
Given an isosceles triangle, the angles opposite the equal sides are equal
Isosceles Triangle Theorem Isosceles Triangle Theorem ConverseConverse
ConclusionAB = AC
A
B C
Given an isosceles triangle, the sides opposite the equal angles are equal
Parallel Line TheoremParallel Line Theorem
1
43
dca b
2
Conclusion
<4 = < a
< 3 < b
<1 < a
<2 = <b
<3 = < c
<4 = <d
<3 + <a = 180°
<4 + <b = 180°
Note: The converse is true also to prove // lines
Sometimes called Corresponding angles
Parallelogram Theorem and Parallelogram Theorem and ConverseConverse
A
CB
DConclusion:
AD = BCAB = DC
BX = XD
AX = XC
x
Opposite Sides
< BAD = <DCB
< ABC = < ADC
Opposite Angles
Diagonals Bisected
In a parallelogram opposite sides are equal, opposite angles are equal and the diagonals bisect each other
Triangle Parallel Similarity Triangle Parallel Similarity TheoremTheorem
A
D
CB
E
Conc
ABC ˜ ADE
Do #3
Test QuestionTest Question
If ABC ˜ XYZ and then < XYZ If ABC ˜ XYZ and then < XYZ is 50°, how much is angle ABC?is 50°, how much is angle ABC?
50°50° Vertically opposite angles is an Vertically opposite angles is an
example of a a) Theorum b) example of a a) Theorum b) axiom c) definition d) postulateaxiom c) definition d) postulate
Pythagoras TheoremPythagoras Theorem
CB
A
a
c b
Given a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides
HYP: Diagram
CONC: b2 = a2 + c2
Pythagoras ExamplesPythagoras Examples
Solve for xSolve for x Solve for xSolve for x
6
8
x
x2 = 62 + 82
x2 = 36 + 64
x = 1020
x
x
202 = x2 + x2
400 = 2x2
200 = x2
14.14 = x2 x = 14.14
The 30-60-90 TheoremThe 30-60-90 Theorem
The side opposite the 30° angle is half the hypotenuse.
HYP: Diagram
CONC: c = ½ b
OR
b = 2c
A
30°
bc
B C
60°
The 30-60-90 Theorem The 30-60-90 Theorem ConverseConverse
If the hypotenuse is twice the length of one of the legs, the angle opposite the leg is 30°
HYP: Diagram
CONC: <ACB = 30°
A
2b
b
B C
30-60-90 Examples 30-60-90 Examples
30°
x6
Opposite the 30°
It is half the hypotenuse
x = 12
30°
14
x
(2x)2=x2+196
4x2=x2+196
3x2=196
x2= 65.33
x = 8.08
Exam Questions Con’tExam Questions Con’t Fill in the missing authoritiesFill in the missing authorities
Statement
Authorities< DAC = <ACB
< DCA = <BAC
AC = AC
Thus DAC BCA
<ABC = < ADC
// Line Theorum
// Line Theorum
Reflex
ASA
Definition
Prove the followingProve the following
HYP: diagram
CONC: AB2 = BC • BD
Statement Authorities<
BAD=<ACDHYP
< ABC = <ABD ABD˜ CBA
AB = BD = AD
CB BA CA
AAReflex
AB2 = BC • BD
DEFN
Cross Multipln
B DC
A
Do #5 & 6
Tips for SuccessTips for Success
Always work on what you knowAlways work on what you know The more facts you put into a The more facts you put into a
question the closer you will get to the question the closer you will get to the answeranswer
Extend the lines Extend the lines