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 formidpoint, distance

Definition ofbisector

January 30,2012Today’s Goals…

Chapter 1 – Tools of Geometry

1.1 Patterns and Inductive Reasoning

1.2 Points, Lines, and Planes

1.3 Segments, Rays, Parallel Lines and Planes

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.

a.) 3, 6, 12, 24… 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.

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

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

5 x 7 = 35

5 x 13 = 65

5 x 3 = 15

5 x 11 = 55

5 x 9 = 45

5 x 25 = 125

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.

P

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.

AB

Planes 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.

PA B C

Plane P Plane ABC

Which points make the plane?

A postulate or axiom is an accepted statement of fact.

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.

We believe it is true just because Euclid said so.

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.

AB

Ray 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.

AB

SRQ

Parallel lines are coplanar lines that do not intersect.

These symbols indicate lines a and b are parallel.

a

b

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

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