1-1 Getting Started1-1 Getting Started

• PointsPoints

- represented by dots- represented by dots

- capital letters for names (A, B, - capital letters for names (A, B, C….etc.)C….etc.)

. B. B

LinesLines• Straight and made up of points Straight and made up of points • A position in space with neither length A position in space with neither length

nor widthnor width• Number linesNumber lines - a numerical value - a numerical value

assigned to each point and the number assigned to each point and the number is its is its coordinatecoordinate

• Extend infinitely far in Extend infinitely far in bothboth directions directions• Arrows on the ends show that the lines Arrows on the ends show that the lines

extend infinitely far in both directionsextend infinitely far in both directions• Lines are given a name or named in Lines are given a name or named in

terms of any two points on the lineterms of any two points on the line

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Line SegmentsLine Segments

• Like lines, segments are made up of Like lines, segments are made up of points and are straightpoints and are straight

• A segment, however, has a definite A segment, however, has a definite beginning and endbeginning and end

• Named in terms of its two endpointsNamed in terms of its two endpoints• Either endpoint can be first in the Either endpoint can be first in the

namename

AB

RaysRays

• Like lines and segments, rays are Like lines and segments, rays are straight and made up of pointsstraight and made up of points

• Begins at an endpoint and then Begins at an endpoint and then extends infinitely far in extends infinitely far in only oneonly one directiondirection

• Must name the endpoint firstMust name the endpoint first

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DefinitionsDefinitions

• Angles are two rays sharing a common endpointAngles are two rays sharing a common endpoint• Triangles are three line segments sharing Triangles are three line segments sharing

common endpointscommon endpoints• Union (U)-joining sets togetherUnion (U)-joining sets together• Intersection ( ) -finding the elements that sets Intersection ( ) -finding the elements that sets

share in commonshare in common

Note: The intersection of any two sides of a Note: The intersection of any two sides of a triangle istriangle is

its vertex. The common endpoint of an its vertex. The common endpoint of an angle is angle is

its vertex its vertex

A B C

D

EF

G

H

IJ

K

Group ActivityGroup Activity

1-2 Measuring Segments1-2 Measuring Segments

• Instruments: ruler, meterstickInstruments: ruler, meterstick• Subtract the coordinate of the Subtract the coordinate of the

smaller endpoint from the larger smaller endpoint from the larger endpointendpoint

• To indicate the measure of a To indicate the measure of a segment, we write the name without segment, we write the name without a line abovea line above

• If P is -4 and Q is 2, find PQ (2 - -4) If P is -4 and Q is 2, find PQ (2 - -4) = 6= 6 0 1 2 3 4 5 6-1-2-3-4-5-6

Measuring Angles Measuring Angles

• Instrument: protractorInstrument: protractor• Measured in degrees (radians, Measured in degrees (radians,

grads)grads)• Amount of turning from the vertexAmount of turning from the vertex

Measuring DegreesMeasuring Degrees

• Each degree of an angle is divided Each degree of an angle is divided into 60 minutesinto 60 minutes

• Each minute of an angle is divided Each minute of an angle is divided into 60 secondsinto 60 seconds

• 1 degree = 60 minutes1 degree = 60 minutes• 1 minute = 60 seconds 1 minute = 60 seconds • Base 60Base 60• Written :Written :45 15 22' "

ConvertConvert

to degrees, minutes, and to degrees, minutes, and

secondsseconds

59 59 60' "

60

Types of AnglesTypes of Angles

• Types of AnglesTypes of Angles– acute: acute:

– right:right:– obtuse:obtuse:– straight:straight:

0 angle measure <90

angle measure = 90

90 angle measure <180

angle measure = 180

• DefinitionDefinition– congruent: having the same shape congruent: having the same shape

or the same size or the same size – diagrams are labeled with “tick marks”diagrams are labeled with “tick marks”

1-3 Collinearity, 1-3 Collinearity, Betweenness, and Betweenness, and

AssumptionsAssumptions• DefinitionsDefinitions

CollinearCollinear: points that are on the same line.: points that are on the same line.

Non-collinearNon-collinear: points that are not on the : points that are not on the same line.same line.

• PropertiesProperties

BetweennessBetweenness: point B is between points A : point B is between points A and C if and only if (IF)and C if and only if (IF)

Triangle Inequality:Triangle Inequality: The sum of the lengths The sum of the lengths of any two sides of a triangle is always of any two sides of a triangle is always greater than the third side.greater than the third side.

AB BC AC

AssumptionsAssumptions

• What is an assumption? What is an assumption? • Assumptions we can make:Assumptions we can make:

– Straight lines are straightStraight lines are straight– Collinearity of pointsCollinearity of points– Betweenness of pointsBetweenness of points

How do we catch ourselves making How do we catch ourselves making assumptions? assumptions?

How do we correct assumptions?How do we correct assumptions?

1-4 Beginning Proofs1-4 Beginning ProofsIs STR a right angle ?

E T R

S

120 3x 60 3x

AnswerAnswer

• Maybe yes, maybe noMaybe yes, maybe no• Since x can be any real number, it is Since x can be any real number, it is

a right angle if and only if x=10. a right angle if and only if x=10.

1-4 Beginning Proofs1-4 Beginning Proofs• ProofProof – a logical argument that shows – a logical argument that shows

a a

statement is true (the validation of a statement is true (the validation of a proposition by application of specified proposition by application of specified rules) rules)

• StatementsStatements – a list of steps ending – a list of steps ending with the conclusionwith the conclusion

• ReasonsReasons – given facts, allowed – given facts, allowed assumptions, definitions, properties assumptions, definitions, properties (i.e. addition, subtraction), theorems(i.e. addition, subtraction), theorems

DefinitionsDefinitions• TheoremTheorem: a mathematical statement : a mathematical statement

that can be provedthat can be proved

• Right Angle TheoremRight Angle Theorem– If two angles are right angles, then they are If two angles are right angles, then they are

congruent.congruent.

• Straight Angle TheoremStraight Angle Theorem– If two angles are straight angles, then If two angles are straight angles, then

they are congruent.they are congruent.

Beginning Proofs: Steps Beginning Proofs: Steps 1.1. Draw a complete and well-labeled diagramDraw a complete and well-labeled diagram2.2. Copy the statement of the problemCopy the statement of the problem3.3. Set up the statements and reasons Set up the statements and reasons

columnscolumns Statements ReasonsStatements Reasons 1. 1.1. 1. 2. 2.2. 2. 3. 3.3. 3. 4. 4. etc. 4. 4. etc. 44. Work with one given statement at a time. Work with one given statement at a time55. Prove the conclusion. Prove the conclusion

1-5 Division of Segments 1-5 Division of Segments and Anglesand Angles

• Segment BisectorSegment Bisector: a point, line, line : a point, line, line segment, or ray that divides a segment segment, or ray that divides a segment into two congruent segments.into two congruent segments.

• MidpointMidpoint: a point that divides, or : a point that divides, or

bisects, a segment into two congruent bisects, a segment into two congruent parts (Rays and lines do not have parts (Rays and lines do not have midpoints. WHY?)midpoints. WHY?)

bis AB CDA B

C

D

G

1-5 Division of Segments 1-5 Division of Segments and Anglesand Angles

Angle BisectorAngle Bisector: a line, line segment, or ray that: a line, line segment, or ray that

divides an angle into two congruent rays.divides an angle into two congruent rays.

PS

T

Q

bis QS PQT��������������

1-5 Division of Segments 1-5 Division of Segments and Anglesand Angles

TrisectTrisect: to divide a line segment or angle : to divide a line segment or angle into threeinto three

congruent partscongruent parts

Trisection pointsTrisection points: the two points at : the two points at which a segmentwhich a segment

is dividedis divided

1-5 Division of Segments 1-5 Division of Segments and Anglesand Angles

TrisectorsTrisectors: lines, line segments, or : lines, line segments, or rays that dividerays that divide

an angle into three congruent parts. an angle into three congruent parts.

AB

C

D

E

1-6 Paragraph Proofs1-6 Paragraph Proofs

• DefinitionDefinition– A written proof in paragraph form using A written proof in paragraph form using

formal mathematical language and logicformal mathematical language and logic

• MethodsMethods– Proving your case directlyProving your case directly– Proving by counterexampleProving by counterexample

• showing that it is impossible to get a true showing that it is impossible to get a true statementstatement

1-7 Deductive Structure1-7 Deductive Structure

• DefinitionDefinition– A system of thought where conclusions A system of thought where conclusions

are justified by previously assumed are justified by previously assumed (what are we allowed to assume?) or (what are we allowed to assume?) or proved statementsproved statements

– The system contains:The system contains:• Undefined termsUndefined terms• Assumptions known as PostulatesAssumptions known as Postulates• DefinitionsDefinitions• Theorems and other conclusionsTheorems and other conclusions

1-7 Deductive Structure1-7 Deductive Structure• Undefined termsUndefined terms: examples are point : examples are point

and lineand line• PostulatePostulate: an unproved assumption: an unproved assumption• DefinitionDefinition: a sentence that states the : a sentence that states the

meaning of a term or ideameaning of a term or idea• They should be reversible They should be reversible • Conditional Statements or ImplicationsConditional Statements or Implications

– “ “ if p, then q”if p, then q”– Can be symbolized p q (read “p implies q”)Can be symbolized p q (read “p implies q”)

• Converse if q, then pConverse if q, then p

• TheoremTheorem: a mathematical statement that : a mathematical statement that can be proved. ( may not be reversible) can be proved. ( may not be reversible)

1-8 Statements of Logic1-8 Statements of Logic• Declarative SentenceDeclarative Sentence “ “Two straight angles are congruent.”Two straight angles are congruent.”

• Conditional StatementConditional Statement “ “If two angles are straight angles, then they are If two angles are straight angles, then they are

congruent.”congruent.” “ “ if p, then q” - p is the hypothesis and q is the if p, then q” - p is the hypothesis and q is the

conclusionconclusion

• Negation Negation – negate a statement– negate a statement - not p- not p - Symbol for not p is ~p- Symbol for not p is ~p - not not p = p, or ~~p=p - not not p = p, or ~~p=p

1-8 Statements of Logic1-8 Statements of Logic

• Every conditional statement “if p, Every conditional statement “if p, then q” has three other statements then q” has three other statements associated with it.associated with it.

ConverseConverse: “if q, then p”: “if q, then p”

Inverse: Inverse: “if not p, then not q”, or “if not p, then not q”, or

“ “if ~p, then ~q”if ~p, then ~q”

Contrapositive: Contrapositive: “if not q, then not “if not q, then not p”, orp”, or

“ “if ~q, then ~p”if ~q, then ~p”

1-8 Statements of Logic1-8 Statements of Logic• Venn DiagramVenn Diagram – a diagram that shows – a diagram that shows

membership in a set or group of setsmembership in a set or group of sets• Chain RuleChain Rule – a way of linking – a way of linking

conditional sets togetherconditional sets together - The conclusion of one statement - The conclusion of one statement

must be the must be the hypothesis of the next statement.hypothesis of the next statement. - “if p, then q”- “if p, then q” “ “if q, then r” if q, then r” “ “if r, then s”if r, then s” “ “if p, then s” if p, then s”

1-9 Probability1-9 Probability• Definition - a ratio describing the likelihood Definition - a ratio describing the likelihood

of something occurring.of something occurring.• Two Basic Steps for Solving ProblemsTwo Basic Steps for Solving Problems

1. Determine all possibilities in a logical 1. Determine all possibilities in a logical

manner. Count them.manner. Count them.

2. Determine the number of these 2. Determine the number of these possibilities possibilities

that are “favorable.” Call these that are “favorable.” Call these winners.winners.

number of winnersProbability =

number of possibilities