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

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

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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?

- Solve the system.
y=x+5

y=-x+7

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 30,2012Today’s Goals…

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

- 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.)

Conjecture

- A conclusion you reach using inductive reasoning.
- A good guess
- The rule you observe

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 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

- 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.

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

- 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.

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.

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

RayA

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

Assignment

- Page 6 1-6, 8,10, 19-24
- Page 13 2-24E

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