Tools of Geometry

1. Tools of Geometry Chapter 1

2. 1-1 Inductive Reasoning • A reasoning based on patterns you observe Example using inductive reasoning: Find the next two terms in the sequence 5, 8, 12, 17, 23, … 2, 6, 18, 54, 162, …

3. Conjecture • A conclusion you make using inductive reasoning Example: Make a conjecture about the sum of the first 8 odd numbers. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 The sum of the first 8 odd numbers form perfect squares.

4. Make a conjecture about the following sequence: 384, 192, 96, 48, … Each term is half the preceding term.

5. Counterexample • An example for which the conjecture is incorrect. Example: Any three points can be connected to form a triangle. Counterexample:

6. Find a counterexample for the following: • All rectangles are squares. 2) The first five counting numbers are prime.

7. 1-3 Points, Lines, and Planes

8. Point – a location A Space – the set of all points Line - a series of points that extends in two opposite directions Collinear Points – Points that lie on the same line

9. Name AB in three other ways. Name a point that is collinear with F. Name a point that is on l and m.

10. Plane • A flat surface that has no thickness containing many lines and extending without end in all directions • Coplanar – points & lines on the same plane

11. Name the plane at the bottom of the cube. Name a line in plane GEF. Name a point coplanar with E and F. Name a line coplanar with EA.

12. Postulate/Axiom – an accepted statement of fact. Postulate 1-1: Through any two points there is exactly one line. Postulate 1-2: If two lines intersect, then they intersect in exactly one point.

13. Postulate 1-3: If two planes intersect, then they intersect in exactly one line. Postulate 1-4: Through any three noncollinear points there is exactly one plane. A C B

14. Different planes in a figure: A B Plane ABCD Plane EFGH Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. D C E F H G

15. Other planes in the same figure: Any three non collinear points determine a plane! Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc.

16. Coplanar Objects Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No

17. 1-4 Segments, Rays, Parallel Lines and Planes

18. Segment: part of a line with two endpoints Ray: Part of a line with only one endpoint Opposite Rays: Two collinear rays with the same endpoint A B C D R E S

19. RAYS ARE DIRECTIONAL!! R Q QR R Q RQ R Q

20. Lines that do not intersect: Parallel: coplanar lines that do not intersect

21. Skew: noncoplanar lines that are not parallel and do not intersect

22. Determine a line that is described by the given: Parallel to BC Skew to GE Skew to GF and coplanar with E Parallel to AE and coplanar with C

23. Parallel planes – Planes that do not intersect Can planes be skew?? Planes can never be skew because they go on continuously in all directions! They will eventually intersect if they are not parallel!

25. 1.5 Measuring Segments: Ruler Postulate: The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers A B x y Coordinates of A and B AB = | x – y |

26. Examples Find: QR RS QS

27. Congruent Segments - two or more segments with the same length GH = RS

29. Determine if the given segments are congruent. LM, MN 3) QN, PL LM, NP 4) LP, NQ

30. Segment Addition Postulate: If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC A B C

31. 1) Determine the length of QS. 2) Suppose QS = 8 cm and QR = 2 cm, find RS.

32. Given: L, M, and N are collinear M is between L and N LM = 3x + 4 MN = 8x – 5 LN = 87 inches Find: x LM MN

33. Given: X, Y, and Z are collinear Y is between X and Z XY = 2x YZ = 5x - 8 XZ = 6x + 6 Find: x XY YZ XZ

34. Given: Find: x GH HI GI

35. H is a midpoint. Midpoint: point that divides a segment into 2 congruent segments A midpoint or line, ray, or segment through a midpoint bisects the segment. Can a segment have more than one midpoint?

37. Angles 1.6 • Two rays with a common endpoint • The rays are the “sides” of the angle and the endpoint is the VERTEX. Side Vertex Side

38. Naming Angles • By the Vertex • B • Using three letters • The letters must go in order • The vertex should be the middle letter ABC or CBA • Using a number in the interior of the angle.

39. Protractor Postulate For every angle, there corresponds a number between 0 and 180. This number is called the "measure" of the angle.

40. Types of Angles Acute Right Obtuse Straight

41. Congruent Angles • Angles with the same measure

42. Angle Addition Postulate • If point B is in the interior of , then A A B B O O C C

43. Angle Addition Postulate (cont.):

44. Identifying Pairs of Angles • Vertical angles – • Two angles whose sides are opposite rays • Adjacent angles – • Two coplanar angles with a common side, a common vertex, and no common interior points. • Complementary angles – • Two angles whose measures have the sum of 90 • Supplementary angles – • Two angles whose measures have the sum of 180

45. In the diagram below, identify: • Adjacent angles • Complementary angles • Supplementary angles • Vertical angles

46. Warm Up: Find: 1) 2) 3) 4)

47. / A and / B are supplements. If m/ A is 22º, what is the m/ B? / A and / B are complements. If / B is twice the measure of / A , what is the measure of / B?

48. Let / A and / B be complementary angles and let m / A = (2x2 + 35)° and m/ B = (x + 10)°. a) What are the values of x? b) What are the measures of the angles?

49. 1-7 Angle and Perpendicular Bisectors

50. Perpendicular lines -Two lines that intersect to form right angles A C D B