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GEOMETRY

LANGUAGE OF. GEOMETRY. UNDEFINED TERMS Points, Lines and Planes The terms points, lines, and planes are the foundations of geometry, but… point, line, and plane are all what we call undefined terms . How can that be?

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GEOMETRY

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  1. LANGUAGE OF GEOMETRY

  2. UNDEFINED TERMS • Points, Lines and Planes • The terms points, lines, and planes are the foundations of geometry, but… • point, line, and plane are all what we call undefined terms. • How can that be? • Any definition we could give them would depend on the definition of some other mathematical idea that these three terms help define. In other words, the definition would be circular!

  3. POINT

  4. Point • First undefined term • No size and no dimension • Merely a position • A dot named with a capital letter A

  5. LINE

  6. AB k k A B Line • Second undefined term • Consist of infinite number of points extending without end in both directions • Usually named with any two of its points or a lower case letter

  7. PLANE

  8. Plane • Third undefined term • Represent a flat surface with no thickness that extends without end in all directions • Usually named by a capital letter or by three points that are not on the same line

  9. Plane Q W R E plane Q or plane EWR

  10. DEFINED TERMS Space Collinear Coplanar Line segment Ray Angles Intersection Congruent

  11. SPACE is the set of all points.

  12. Space • Space is the set of all points

  13. Space • At least four noncoplanar points distinguish space. B D A C

  14. Collinear points are points that lie on the same line.

  15. Collinear Points • Points are collinear if and only if they lie on the same line. • Points are collinear if they lie on the same line • Points on the same line are collinear.

  16. Collinear Points • C, A and B are collinear. D C A B

  17. Collinear Points • Points that are not collinear are noncollinear. D C A B

  18. Collinear Points • D, A and B are noncollinear. D C A B

  19. Coplanar points are points that lie on the same plane.

  20. Coplanar Points • Points are coplanar if and only if they lie on the same plane. • Points are coplanar if they lie on the same plane. • Points lie on the same plane are coplanar.

  21. Coplanar Points • E, U, W and R are coplanar • T, U, W and R are noncoplanar T U W R E

  22. Here is line AB, or recall symbolically The line segment does not extend without end. It has endpoints, in this case A and B. The segment contains all the points on the line between A and B A B A B Notice the difference in the symbolic notation! Line Segment • Let’s look at the idea of a point in between two other points on a line. This is segment

  23. A is called the initial point The initial point is always the first letter in naming a ray. Notice the difference in symbols from both a line and segment. A B Ray AB extends in one direction without end. Ray • Symbolized by Let’s look at a ray:

  24. is the same as A B A B Symbol alert! • Not all symbols are created equal! is the same as BUT…

  25. Initial point 1st is not the same as A B A B Symbol alert!! The ray is different! Notice that the initial point is listed first in the symbol. Also note that the symbolic ray always has the arrowhead on the right regardless of the direction of the ray.

  26. A B C Opposite Rays • If C is between A and B, then and are opposite rays. C is the common initial point for the rays!

  27. Angles • Rays are important because they help us define something very important in geometry…Angles! • An angle consists of two different rays that have the same initial point. The rays are sides of the angles. The initial point is called the vertex. Notation: We denote an angle with three points and symbol. The middle point is always the vertex. We can also name the angle with just the vertex point. This angle can be denoted as: B vertex sides A C

  28. A A A A Classifying Angles • Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less or equal to 180°. Acute angle 0°< m A < 90° Right angle m A = 90° Obtuse angle 90°< m A < 180° Straight angle m A = 180°

  29. Intersection is the set of points common to two or more figures.

  30. Intersections of lines and planes • Two or more geometric figures intersect if they have one or more points in common. • The intersection of the figures is the set of points the figure has in common How do 2 line intersect? How do 2 planes intersect? What about a line and a plane? Think!!

  31. Intersection k C A B Line k intersects CB at A

  32. B A F D Modeling Intersections • To think about the questions on the last slide lets look at the following… Point E is the intersection of plane H and line EC E Two lines intersect at a point, like here at point A. H C G Line BF is the intersection of the planes G and H.

  33. Intersection • A set of points is the intersection of two figures if and only if the points lie in both figures

  34. What is Congruency? • When two shapes are exactly the same size and shape we say they are CONGRUENT WHICH OF THESE SHAPES IS CONGRUENT WITH THE FIRST SHAPE?

  35. Too small Too big Even though their direction changed, they are the same size and shape.

  36. Which of the following are congruent?

  37. Which of the following are congruent?

  38. Angles and Line segments can be congruent • Two segments are congruent if and only if they have the same measure • Two angles are congruent if and only if they have the same measure • Mark angles and segments that have the same measure with like markings.

  39. Based on the picture, Geometry is telling me something. Share it.

  40. In conveying ideas, what is the advantage of presenting complex concepts in organized fashion with well-defined relationships? Journal

  41. ANGLES

  42. We are going to learn facts about anglesso that we can calculate their size without drawing or measuring • Angles on a straight line • Angles around a point • Angles between intersecting lines

  43. Complementary angles When two angles add up to 90° they are called complementary angles. a b a + b = 90° Angle a and angle b are complementary angles.

  44. Supplementary angles When two angles add up to 180° they are called supplementary angles. b a a + b = 180° Angle a and angle b are supplementary angles.

  45. Angles on a straight line Angles on a line add up to 180. a b a + b = 180° because there are 180° in a half turn.

  46. Angles on a straight line

  47. Intersecting lines

  48. Vertically opposite angles a d b c When two lines intersect, two pairs of vertically opposite angles are formed. and a = c b = d Vertically opposite angles are equal.

  49. Angles around a point Angles around a point add up to 360. b a c d a + b + c + d = 360 because there are 360 in a full turn.

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