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

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.

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  1. 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?

  2. Do you remember? • Solve the system. y=x+5 y=-x+7

  3. 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

  4. January 30,2012Today’s Goals…

  5. 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.

  6. 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.

  7. 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.)

  8. Conjecture • A conclusion you reach using inductive reasoning. • A good guess • The rule you observe

  9. 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

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. Which points make the plane?

  18. 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.

  19. 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.

  20. 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.

  21. a b Parallel lines are coplanar lines that do not intersect. • These symbols indicate lines a and b are parallel. a || b

  22. Skew lines are noncoplanar; therefore, they are not parallel and do not intersect. AB || EF AB and CG are skew.

  23. Parallel planes are planes that do not intersect. Plane ABCD || Plane GHIJ

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

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