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# Warm Up January 30,2012 Please turn in your worksheets. - PowerPoint PPT Presentation

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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Presentation Transcript

• 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?

• Solve the system.

y=x+5

y=-x+7

• Area of

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

• Other Formulas for

midpoint, distance

• Definition of

bisector

January 30,2012Today’s Goals…

• 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.

• 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.

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.)

• A conclusion you reach using inductive reasoning.

• A good guess

• The rule you observe

• 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 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.

• 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 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.

• 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

• 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.

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.

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.

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.

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.

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.

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.

AB || EF

AB and CG are skew.

Parallel planes are planes that do not intersect.

Plane ABCD || Plane GHIJ

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