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Warm Up January 30,2012 Please turn in your worksheets. If ray BD is a bisector of <ABC: a) and m <ABC equals 70 degrees, what is the measure of <BDC? b) and m <ABC equals (x+12) and m <BDC equals (2x-36), what is x?. Do you remember?. Solve the system. y=x+5 y=-x+7.

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Warm up january 30 2012 please turn in your worksheets

Warm Up January 30,2012Please turn in your worksheets.

  • If ray BD is a bisector of <ABC:

    a) and m<ABC equals 70 degrees, what

    is the measure of <BDC?

    b) and m<ABC equals (x+12) and

    m<BDC equals (2x-36), what is x?


Do you remember

Do you remember?

  • Solve the system.

    y=x+5

    y=-x+7


What were the 10 formulas from last week

What were the 10 formulas from last week?

  • Area of

    square, parallelogram, triangle, circle, regular polygon, sector, trapezoid

  • Other Formulas for

    midpoint, distance

  • Definition of

    bisector


January 30 2012 today s goals

January 30,2012Today’s Goals…


Deductive reasoning

Deductive Reasoning

  • Given a rule, state the example belongs.

  • Example: Every square is a rectangle.

  • ABCD is a square so by deductive reasoning ABCD is a rectangle.


Inductive reasoning

Inductive Reasoning

  • Reasoning that is based on patterns you observe.

  • If you observe a pattern in a sequence, you can use inductive reasoning to tell what the next term in the sequence will be.

  • See the examples follow a pattern then write the rule.


Warm up january 30 2012 please turn in your worksheets

a.) 3, 6, 12, 24…

Ex.1: Finding and Using a PatternFind a pattern for each sequence. Use the pattern to show the next two terms in the sequence.

b.)

You Try…

c.) 1, 2, 4, 7, 11, 16, 22, … d.)


Conjecture

Conjecture

  • A conclusion you reach using inductive reasoning.

  • A good guess

  • The rule you observe


Do you see the pattern

Do you see the pattern?

  • State the rule then identify the next two terms.

    1) o,t,t,f,f,s,s,e

    2) Aquarius, Pisces, Aries, Taurus


Ex 2 using inductive reasoning make a conjecture about the sum of the first 30 odd numbers

Ex.2: Using Inductive ReasoningMake a conjecture about the sum of the first 30 odd numbers.

  • What do you notice?

    1=

    1 + 3=

    1 + 3 + 5=

    1 + 3 + 5 + 7=

  • Using inductive reasoning, you can conclude that the sum of the first 30 odd numbers is 302, or 900.


Counterexample

Counterexample

  • Not all conjectures turn out to be true.

  • You can prove that a conjecture is false by finding ONE counterexample.

  • A counterexample to a conjecture is an example for which the conjecture is incorrect.


Ex 3 testing a conjecture some products have 5 as a factor as shown

Ex.3: Testing a ConjectureSome products have 5 as a factor, as shown.

  • Which conjecture is true?

  • If false, state a counterexample.

    • The product of 5 and any odd number is odd.

    • The product of 5 and any number ends in 5.


The beginning of geometric thought

The beginning of geometric thought

  • To start off we have to have some words without a definition. We have an understanding of what they are.

  • The three words are point, line and plane.


Point

P

Point

  • You can think of a point as a location.

    • No size

    • Represented by a small dot

    • Named by a capital letter

  • Space is defined as the set of all points.


Warm up january 30 2012 please turn in your worksheets

A

B

Line

  • You can think of a line as a series of points that extends in two opposite directions without end.

  • Name a line two different ways:

    • Use two points on the line such as AB (read “line AB”)

    • Use a single lowercase letter such as “line t”

  • Collinear points are points that lie on the same line.


Planes

Planes

P

A B

C

Plane P

Plane ABC

  • A plane is a flat surface that has no thickness.

    • A plane contains many lines and extends without end in the direction of all its lines.

    • You can name a plane by either a single capital letter or by at least 3 of its noncollinear points.

  • Points and lines in the same plane are coplanar.


Which points make the plane

Which points make the plane?


A postulate or axiom is an accepted statement of fact

A postulate or axiom is an accepted statement of fact.

We believe it is true just because Euclid said so.

  • The First Three Postulates:

    • Through any two points there is exactly one line.

    • If two lines intersect, then they intersect in exactly one point.

    • If two planes intersect, then they intersect in exactly one line.


Segment

A

B

Segment

  • Many geometric figures, such as squares and angles, are formed by parts of lines called segments or rays.

  • A segment is the part of a line consisting of two endpoints and all points between them.


Warm up january 30 2012 please turn in your worksheets

Q

R

S

Ray

A

B

  • A ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.

  • Opposite rays are two collinear rays with the same endpoint. Opposite rays ALWAYS form a line.


Parallel lines are coplanar lines that do not intersect

a

b

Parallel lines are coplanar lines that do not intersect.

  • These symbols indicate lines a and b are parallel.

a || b


Skew lines are noncoplanar therefore they are not parallel and do not intersect

Skew lines are noncoplanar; therefore, they are not parallel and do not intersect.

AB || EF

AB and CG are skew.


Parallel planes are planes that do not intersect

Parallel planes are planes that do not intersect.

Plane ABCD || Plane GHIJ


Assignment

Assignment

  • Page 6 1-6, 8,10, 19-24

  • Page 13 2-24E


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