Geometry Proofs Geometry Proofs

Math 416Math 416

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

Triangle Sum TheoremTriangle Sum Theorem

1

32

Conclusion<1 + <2 + <3 = 180°

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 QuestionExam Question

A

CB

DHyp: Diagram

Construction AC

Conc: < ABC = < ADC

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

Exam Questions & PracticeExam Questions & Practice

We will do more examples on the We will do more examples on the board together… board together…

P262, p266, 267, 268, p272, 274P262, p266, 267, 268, p272, 274 Study GuideStudy Guide Test Test