1-1 Getting Started PointsPoints - represented by dots - represented by dots - capital letters for names (A, B, C….etc.) - capital letters for names (A, B, C….etc.). B. B

Lines Straight and made up of pointsStraight and made up of points A position in space with neither length nor widthA position in space with neither length nor width Number lines - a numerical value assigned to each point and the number is its coordinateNumber lines - a numerical value assigned to each point and the number is its coordinate Extend infinitely far in both directionsExtend infinitely far in both directions Arrows on the ends show that the lines extend infinitely far in both directionsArrows on the ends show that the lines extend infinitely far in both directions Lines are given a name or named in terms of any two points on the lineLines are given a name or named in terms of any two points on the line

Line Segments Like lines, segments are made up of points and are straightLike lines, segments are made up of points and are straight A segment, however, has a definite beginning and endA segment, however, has a definite beginning and end Named in terms of its two endpointsNamed in terms of its two endpoints Either endpoint can be first in the nameEither endpoint can be first in the name

Rays Like lines and segments, rays are straight and made up of pointsLike lines and segments, rays are straight and made up of points Begins at an endpoint and then extends infinitely far in only one directionBegins at an endpoint and then extends infinitely far in only one direction Must name the endpoint firstMust name the endpoint first

Definitions Angles are two rays sharing a common endpointAngles are two rays sharing a common endpoint Triangles are three line segments sharing common endpointsTriangles are three line segments sharing common endpoints Union (U)-joining sets togetherUnion (U)-joining sets together Intersection ( ) -finding the elements that sets share in commonIntersection ( ) -finding the elements that sets share in common Note: The intersection of any two sides of a triangle is Note: The intersection of any two sides of a triangle is its vertex. The common endpoint of an angle is its vertex. The common endpoint of an angle is its vertex its vertex

Group Activity

1-2 Measuring Segments Instruments: ruler, meterstickInstruments: ruler, meterstick Subtract the coordinate of the smaller endpoint from the larger endpointSubtract the coordinate of the smaller endpoint from the larger endpoint To indicate the measure of a segment, we write the name without a line aboveTo indicate the measure of a segment, we write the name without a line above If P is -4 and Q is 2, find PQ (2 - -4) = 6If P is -4 and Q is 2, find PQ (2 - -4) =

Measuring Angles Instrument: protractorInstrument: protractor Measured in degrees (radians, grads)Measured in degrees (radians, grads) Amount of turning from the vertexAmount of turning from the vertex

Measuring Degrees Each degree of an angle is divided into 60 minutesEach degree of an angle is divided into 60 minutes Each minute of an angle is divided into 60 secondsEach minute of an angle is divided into 60 seconds 1 degree = 60 minutes1 degree = 60 minutes 1 minute = 60 seconds1 minute = 60 seconds Base 60Base 60 Written :Written :

Convert to degrees, minutes, and seconds

Types of Angles Types of AnglesTypes of Angles – acute: – right: – obtuse: – straight: DefinitionDefinition – congruent: having the same shape or the same size or the same size – diagrams are labeled with “tick marks”

1-3 Collinearity, Betweenness, and Assumptions DefinitionsDefinitions Collinear: points that are on the same line. Collinear: points that are on the same line. Non-collinear: points that are not on the same line. Non-collinear: points that are not on the same line. PropertiesProperties Betweenness: point B is between points A and C if and only if (IF) Betweenness: point B is between points A and C if and only if (IF) Triangle Inequality: The sum of the lengths of any two sides of a triangle is always greater than the third side. Triangle Inequality: The sum of the lengths of any two sides of a triangle is always greater than the third side.

Assumptions What is an assumption?What is an assumption? Assumptions we can make:Assumptions we can make: –Straight lines are straight –Collinearity of points –Betweenness of points How do we catch ourselves making assumptions? How do we correct assumptions?

1-4 Beginning Proofs 1-4 Beginning Proofs ETR S

Answer Maybe yes, maybe noMaybe yes, maybe no Since x can be any real number, it is a right angle if and only if x=10.Since x can be any real number, it is a right angle if and only if x=10.

1-4 Beginning Proofs Proof – a logical argument that shows aProof – a logical argument that shows a statement is true (the validation of a proposition by application of specified rules) statement is true (the validation of a proposition by application of specified rules) Statements – a list of steps ending with the conclusionStatements – a list of steps ending with the conclusion Reasons – given facts, allowed assumptions, definitions, properties (i.e. addition, subtraction), theoremsReasons – given facts, allowed assumptions, definitions, properties (i.e. addition, subtraction), theorems

Definitions Theorem: a mathematical statement that can be provedTheorem: a mathematical statement that can be proved Right Angle TheoremRight Angle Theorem – If two angles are right angles, then they are congruent. congruent. Straight Angle TheoremStraight Angle Theorem – If two angles are straight angles, then they are congruent.

Beginning Proofs: Steps 1.Draw a complete and well-labeled diagram 2.Copy the statement of the problem 3.Set up the statements and reasons columns Statements Reasons Statements Reasons etc etc. 4. Work with one given statement at a time 5. Prove the conclusion

1-5 Division of Segments and Angles 1-5 Division of Segments and Angles Segment Bisector: a point, line, line segment, or ray that divides a segment into two congruent segments.Segment Bisector: a point, line, line segment, or ray that divides a segment into two congruent segments. Midpoint: a point that divides, or bisects, a segment into two congruent parts (Rays and lines do not have midpoints. WHY?)Midpoint: a point that divides, or bisects, a segment into two congruent parts (Rays and lines do not have midpoints. WHY?) A B C D G

1-5 Division of Segments and Angles Angle Bisector: a line, line segment, or ray that Angle Bisector: a line, line segment, or ray that divides an angle into two congruent rays. divides an angle into two congruent rays. P S T Q

1-5 Division of Segments and Angles Trisect: to divide a line segment or angle into three congruent parts Trisection points: the two points at which a segment is divided

1-5 Division of Segments and Angles Trisectors: lines, line segments, or rays that divide Trisectors: lines, line segments, or rays that divide an angle into three congruent parts. an angle into three congruent parts. A B C D E

1-6 Paragraph Proofs 1-6 Paragraph Proofs DefinitionDefinition –A written proof in paragraph form using formal mathematical language and logic MethodsMethods – Proving your case directly – Proving by counterexample showing that it is impossible to get a true statementshowing that it is impossible to get a true statement

1-7 Deductive Structure 1-7 Deductive Structure DefinitionDefinition –A system of thought where conclusions are justified by previously assumed (what are we allowed to assume?) or proved statements –The system contains: Undefined termsUndefined terms Assumptions known as PostulatesAssumptions known as Postulates DefinitionsDefinitions Theorems and other conclusionsTheorems and other conclusions

1-7 Deductive Structure Undefined terms: examples are point and lineUndefined terms: examples are point and line Postulate: an unproved assumptionPostulate: an unproved assumption Definition: a sentence that states the meaning of a term or ideaDefinition: a sentence that states the meaning of a term or idea They should be reversibleThey should be reversible Conditional Statements or ImplicationsConditional Statements or Implications –“ if p, then q” –Can be symbolized p q (read “p implies q”) Converse if q, then p Converse if q, then p Theorem: a mathematical statement that can be proved. ( may not be reversible)Theorem: a mathematical statement that can be proved. ( may not be reversible)

1-8 Statements of Logic 1-8 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 congruent.” “If two angles are straight angles, then they are congruent.” “ if p, then q” - p is the hypothesis and q is the conclusion “ if p, then q” - p is the hypothesis and q is the conclusion Negation – negate a statementNegation – 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 Logic Every conditional statement “if p, then q” has three other statements associated with it.Every conditional statement “if p, then q” has three other statements associated with it. Converse: “if q, then p” Converse: “if q, then p” Inverse: “if not p, then not q”, or Inverse: “if not p, then not q”, or “if ~p, then ~q” “if ~p, then ~q” Contrapositive: “if not q, then not p”, or Contrapositive: “if not q, then not p”, or “if ~q, then ~p” “if ~q, then ~p”

1-8 Statements of Logic Venn Diagram – a diagram that shows membership in a set or group of setsVenn Diagram – a diagram that shows membership in a set or group of sets Chain Rule – a way of linking conditional sets togetherChain Rule – a way of linking conditional sets together - The conclusion of one statement must be the - The conclusion of one statement 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 Probability 1-9 Probability Definition - a ratio describing the likelihood of something occurring.Definition - a ratio describing the likelihood 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 possibilities 2. Determine the number of these possibilities that are “favorable.” Call these winners. that are “favorable.” Call these winners.