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Definition of Geometry

Definition of Geometry. The branch of mathematics concerned with properties of and relations between points, lines, planes and figures. Origins Of Geometry.

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Definition of Geometry

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  1. Definition of Geometry The branch of mathematics concerned with properties of and relations between points, lines, planes and figures.

  2. Origins Of Geometry The earliest records of Geometry can be traced to ancient Egypt and the Indus Valley from around 3000 BC. Early Geometry was a collection of observed principles concerning lengths, angles, area, and volumes. These principles were developed to meet practical needs in construction, astronomy, and other needs.

  3. Origins of Geometry Euclid, a Greek mathematician, wrote The Elements of Geometry. It is considered one of the most important early texts on Geometry. He presented geometry in a practical form known as Euclidean geometry. Euclid was not the first elementary Geometry textbook, but the others fell into disuse and were lost.

  4. Chapter 1 Basics of Geometry

  5. 1.1 Patterns and Inductive Reasoning Vocabulary: A conjecture is an unproven statement that is based on observations. Example: 3 + 4 + 5 = 4 x 3 6 + 7 + 8 = 7 x 3 4 + 5 + 6 = 5 x 3 7 + 8 + 9 = 8 x 3 5 + 6 + 7 = 6 x 3 8 + 9 + 10 = 9 x 3 Conjecture: The sum of any three consecutive integers is 3 times themiddle number.

  6. Example: Conjecture: The sum of any two odd numbers is ____. 1 + 1 = 2 1 + 3 = 4 3 + 5 = 8 7 + 11 = 18 13 + 19 = 32

  7. Inductive reasoning is a process that includes looking for patterns and making conjectures. Prediction or conclusion based on observation of a pattern.

  8. Example—expecting a traffic light to stay green for 40 seconds because you have seen it stay green for 40 seconds many times before. Example—When you see the numbers 2, 4, 6, 8, and 10, and you expect the next number to be 12. Predict the next number. 2, 6, 18, 54, …

  9. A counterexample is an example that shows a conjecture is false. Example: Show the conjecture is false by finding a counterexample: The difference of two positive numbers is always positive. Answer: 3 – 9 = -6

  10. Describe pattern and predict next number • 1. 128, 64, 32, 16, … • 2. 5, 4, 2, -1, …

  11. Example • Given the pattern __, -6, 12, __, 48, … answer the following exercises: a. Fill in the missing numbers. b. Determine the next two numbers in this sequence. c. Describe how you determined what numbers completed the sequence. d. Are there any other numbers that would complete this sequence?

  12. Answer to previous question. • A) 3, -6, 12, -24, 48 • B) the next two numbers are -96 and 192 • D) -24, -6, 12, 30, 48 next two numbers: 66 and 84

  13. Complete the Conjecture • The sum of the first n even positive integers is _____? 1st even integer: 2 = 1(2) Sum of 1st two even pos. integers: 2 + 4 = 6 = 2(3) Sum of 1st three…: 2 + 4 + 6 = 12=3(4) Sum of 1st four…: 2 + 4 + 6 + 8 = 20= 4(5) Sum of 1st n even pos. Int. is: n(n + 1)

  14. Counterexample • Show the conjecture is false by finding a counterexample. • Conjecture: If the difference of two numbers is odd, then the greater of the two numbers must also be odd.

  15. Finding the nth term • http://www.math-magic.com/sequences/nth_term.htm

  16. 1.2 Points, Lines, Planes • The three undefined terms are point, line, and plane. • A point has no dimension. • A point is usually named by a capital letter. ٭ A All geometric figures consist of points.

  17. Lines • Lines extend indefinitely and have no thickness or width. • Lines are usually named by lower case script letters or by writing capital letters for 2 points on the line, with a double arrow over the pair of letters. l line l A B C AB BC AC

  18. Plane • A plane extends in two dimensions. It is usually represented by a shape that looks like a tabletop or wall. • A plane is a flat surface that extends indefinitely in all directions. • A plane can be named by a capital script letter or by three noncollinear points in the plane. plane ABC or plane R • http://www.mathopenref.com/plane.html

  19. Space-Three dimensional set of all points. • Collinear points are points that lie on the same line. • Coplanar points are points that lie on the same plane. A

  20. Line Segment, Endpoint • A line segment is part of a line that consists of two points, called endpoints, and all points on the line between the endpoints. Name by using endpoints. • B AB A

  21. Ray, Initial point A ray is part of a line that consists of a point, called an initial point, and all points on the line that extend in one direction. Name by using the endpoint first, then any point of ray. A B AB

  22. Opposite Rays • If C is between A and B on AB, then CA and CB are opposite rays. B C A

  23. Two or more geometric figures intersect if they have one or more points in common. The intersection of two or more geometric figures is the set of points that the figures have in common.

  24. Homework Pages 6 – 9 #’s 12 – 15, 16 – 22 evens, 25 30, 47, 48 Pages 13 – 16 #’s 10 – 16 evens, 25 – 31 odds, 37, 44 - 47

  25. 1.3 Segments and their Measures • Postulates are rules that are accepted without proof. Postulates are also called axioms. Ex: A line contains at least two points. • A coordinate is a real number that corresponds to a point on a line. • The distance between two points on a line is the absolute value of the difference between the coordinates of the points.

  26. The length of a segment is the distance between the endpoints. • When three points lie on a line, you can say that one of them is between the other two. • The Distance Formula is a formula for finding the distance between two points in a coordinate plane. • Congruent segments are segments that have the same length.

  27. Postulate 1: Ruler Postulate • The points on a line can be matched one to one with real numbers. The real number that corresponds to a point is the coordinate of the point. • The distance between points A and B, written AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the length of AB.

  28. Finding distance between two points • AB =

  29. Find the distance between the points • 12) E and A • 13) F and B • 14) E and D • 15) C and B • 16) F and A

  30. 2 2 Distance Formula: Given the two points (x1, y1) and (x2, y2), the distance between these points is given by the formula: d =

  31. Example B (3, 4) A(-1,2) D(-3,-2) C(4, -3) Find AB, BC, CD, and AD.

  32. Postulate 2: Segment Addition Postulate • If B is between A and C, then AB + BC = AC. • If AB + BC = AC, then B is between A and C.

  33. A lies between C and T. Find CT if CA is 5 and AT is 8. • Find AC if CT is 20 and AT is 8. • See Notetaking guide book.

  34. 1.4 Angles and Their Measures • An angle consists of two noncollinear rays that have the same initial point. A < ABC with sides BA and BC Name the angle: <ABC or <CBA or <B or <1 B 1 The initial point of the rays is the vertex of the angle. The vertex is point B. C

  35. Congruent angles • Congruent angles are angles that have the same measure. m<A = m<B B A

  36. Measure of an angle • In <AOB, ray OA and ray OB can be matched one to one with the real numbers from 0 to 180. • The measure of <AOB is equal to the absolute value of the difference between the real numbers for ray OA and ray OB.

  37. Acute Angle • An acute angle is an angle that measures between 0° and 90°. http://optics.org/cws/article/research/23663

  38. Angles • An angle separates a plane into three parts: the interior, the exterior, and the angle itself. exterior interior

  39. Right Angle • A right angle is an angle that measures 90°.

  40. Obtuse Angle • An obtuse angle is an angle that measures between 90° and 180°.

  41. Straight Angle • A straight angle is an angle that measures 180º.

  42. Adjacent Angles • Two angles are adjacent if they have a common vertex and side, but have no common interior points. D A D B C B C A <ABD AND <DBC are adjacent <ABD AND <DBC are adjacent

  43. Adjacent Angles 1 4 2 3 Adjacent Angles: <1 and <2; <2 and <3; <3 and <4; <4 and <1 NOTE: Not adjacent <1 and <3, <4 and <2

  44. Name the angles in the figure K M J L

  45. Protractor Postulate • For every angle there is a unique real number r, called its degree measure, such that 0 < r < 180. *ILLUSTRATION ON NEXT SLIDE

  46. A B A O

  47. Angle Addition Postulate • If P is in the interior of <RST, then m<RSP + m<PST = m<RST R P S T

  48. Measure the angle. Then classify the angle as acute, right, obtuse, or straight. • <AFD • <AFE • <BFD • <BFC

  49. Homework Pages 21 – 24 #’s 20, 24 – 30 evens, 31, 33, 35, 48, 55, 56 Pages 29 – 32 #’s 14 – 34 evens, 68, 70 - 73

  50. 1.5 Segment and Angle Bisector • A midpoint is the point that divides, or bisects, a segment into two congruent segments. • The midpoint M of PQ is the point between P and Q such that PM = MQ. M Q P

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