Math 310

1. Math 310 Section 9.1 Geometry Introduction

2. Axiomatic System A logical system which possesses an explicitly stated set of axioms from which theorems can be derived. (mathworld.wolfram.com)

3. www.xkcd.com

4. Undefined Terms • Point • Line • Plane

5. “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” - David Hilbert

6. Linear “Notions” • Collinear points • Is between • Line segment (segment) • Ray

7. Collinear points (& non collinear)

8. Is Between

9. Line Segment

10. Ray

11. Planar “Notions” • Coplanar points • Noncoplanar points • Coplanar lines • Skew lines • Intersecting lines • Concurrent lines • Parallel lines

12. Coplanar & Noncoplanar Points

13. Coplanar lines

14. Skew Lines

15. Intersecting Lines

16. Concurrent Lines

17. Parallel Lines

18. Properties of “tables, chairs and beer mugs” • There is exactly one line that contains any two distinct points • If two points lie in a plane, then the line containing the points lies in the plane. • If two distinct planes intersect, then their intersection is a line. • There is exactly one plane that contains any three distinct noncollinear points.

19. Properties (cont) • A line and a point not on the line determine a plane. • Two parallel lines determine a plane • Two intersecting lines determine a plane.

20. B A A B Property 1

21. Property 2

22. Property 3

23. Property 4

24. Property 5

25. Property 6

26. Property 7

27. Intersecting Planes • Parallel • Along a line

28. Parallel Planes

29. Planes Intersecting along a line

30. Angle, Vertex, Side Def When two rays share an endpoint, an angle is formed. The common initial point of the rays is the vertex of the angle. Each ray is called a side of the angle.

31. Angle: <ABC Side Vertex Ex.

32. Ex. <EBF <GFE <I

33. Ex. <MJK <NKL <OLJ <KJL <LKJ <JLK Name all six angles.

34. Angle Measure To measure an angle we use the unit degree. It measures the “opening” of the angle. The largest angle measure is 360° and the smallest is 0°. A complete rotation about a point is 360°. For more accuracy, angles can be further measure in minutes, and seconds. Each degree is divided into 60 minutes, and each minute is divided into 60 seconds.

35. Ex. Add: 45°23’47” and 62°36’51” 45°23’47” + 62°36’51” = 108°0’38” or 108°38” Add: 145°17’4” and 220°31’32” 145°17’4” + 220°31’32” = 365°48’36” = 5°48’36”

36. Ex. Solve for x. m<ABC = 80° m<ABD = 30° m<DBC = (x – 25)° x = 75 m<ABC = 82° m<ABD = (x – 13)° m<DBC = (x + 7)° x = 44

37. Protractor A protractor is a standard tool for measuring angles. To use, line the vertex up with the center of the base of the protractor and line one side of the angle up with the 0° mark. Now measure from 0°, increasing, until you see the other side of the angle and read the mark.

38. Types of Angles • Obtuse • Acute • Right • Straight

39. Ex. acute obtuse straight right

40. Perpendicular Lines Def. Two lines are perpendicular if they intersect and form right angles. perpendicular not perpendicular

41. Line Perpendicular to a Plane A line is perpendicular to a plane if it is perpendicular to every line, contained in the plane, passing through the point of intersection.

42. Ex.

43. Ex.

44. Questions • Is it possible for a line intersecting a plane to be perpendicular to exactly one line in the plane through its intersection with the plane? • Can a line intersecting a plane be perpendicular to exactly two distinct lines in the plane going through the point of intersection? • Yes • No

45. Line Perpendicular to a Plane Thrm A line perpendicular to two distinct lines in the plane through its intersection with the plane is perpendicular to the plane.