1. Review of Geometry Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT TOPICSBACKNEXT © 2002 East Los Angeles College. All rights reserved. Click one of the buttons below or press the enter key

2. Topics Lines Angles Triangles Click on the topic that you wish to view... EXIT TOPICSBACKNEXT

3. Lines EXIT TOPICSBACKNEXT

4. When a pair of lines are drawn, the portion of the plane where the lines do not intersect is divided into three distinct regions. Region 1 Region 3 Region 2 EXIT TOPICSBACKNEXT

5. These regions are referred to as: Interior Region – Region bounded by both lines. Exterior Region – The remaining outside regions. exterior interior EXIT TOPICSBACKNEXT

6. Parallel Lines – Lines that never intersect. l1l1 l2l2 Notation l 1 l 2 EXIT TOPICSBACKNEXT

7. Transversal – A line that intersects two or more lines in different points. l1l1 l2l2 Note: l 1 is not parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT

8. Transversal l1l1 l2l2 Note: l 1 is parallel to l 2 ( l 1 l 2 ) EXIT TOPICSBACKNEXT

9. Angles EXIT TOPICSBACKNEXT

10. Angles are formed when lines intersect. l1l1 l2l2 Note: ( l 1 l 2 ) A B C D EXIT TOPICSBACKNEXT

11.  A and  B are said to be adjacent. (neighbors) l1l1 l2l2 A B C D EXIT TOPICSBACKNEXT

12. l1l1 l2l2 A B C D Adjacent Angles – Angles that share a common vertex and a common side between them. EXIT TOPICSBACKNEXT

13. l1l1 l2l2 A B C D Note:  B and  C are adjacent (neighbors)  C and  D are adjacent (neighbors)  D and  A are adjacent (neighbors) EXIT TOPICSBACKNEXT

14. l1l1 l2l2 A B C D Vertical Angles – The pairs of non-adjacent angles formed by the intersection of two lines. EXIT TOPICSBACKNEXT

15. l1l1 l2l2 A B C D Note:  A and  C are vertical angles  B and  D are vertical angles EXIT TOPICSBACKNEXT

16. Q: What’s special about vertical angles? Answer – They have the same measure. (they are congruent) l1l1 l2l2 110° 70° EXIT TOPICSBACKNEXT

17. Fact – When you intersect two lines at a point l1l1 l2l2 A C BD  A   C (congruent)  B   D (congruent) EXIT TOPICSBACKNEXT

18. Two angles are said to be supplementary if their sum measures 180°. Adjacent angles formed by two intersecting lines are supplementary. l1l1 l2l2 A C BD  A and  B are supplementary angles. EXIT TOPICSBACKNEXT

19. Can you find any other supplementary angles in the figure below? l1l1 l2l2 A C BD EXIT TOPICSBACKNEXT

20. Note: Angles whose sum measures 90° are said to be complementary. EXIT TOPICSBACKNEXT

21. Revisiting the transversal, copy this picture in your notebook. l1l1 l2l2 Note: ( l 1 l 2 ) AB C D H G E F EXIT TOPICSBACKNEXT

22. Angles in the interior region between the two lines are called interior angles. Angles in the exterior region are called exterior angles. l1l1 l2l2 AB C D H G E F Interior Exterior EXIT TOPICSBACKNEXT

23. Q: Which are the interior angles and exterior angles? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT

24. l1l1 l2l2 AB C D H G E F Answer— InteriorExterior  C  A  D  B  E  G  F  H EXIT TOPICSBACKNEXT

25. Q: Which angles are adjacent? Q: Which angles are vertical? Q: Which angles are supplementary? l1l1 l2l2 AB C D H G E F EXIT TOPICSBACKNEXT

26. Consider a transversal consisting of the two parallel lines. l1l1 l2l2 A C B D FE GH EXIT TOPICSBACKNEXT

27. l1l1 l2l2 A C B D FE GH We know,  A   D  B   C  E   H  G   F since they are all vertical angles. EXIT TOPICSBACKNEXT

28. Q: Are any other angles congruent? EXIT TOPICSBACKNEXT

29. Yes! If we could slide l 2 up to l 1, we would be looking at the following picture. EXIT TOPICSBACKNEXT

30. l1l1 l2l2 A C B D FE GH This means the following is true:  A and  E have the same measure (congruent)  B and  F have the same measure (congruent)  C and  G have the same measure (congruent)  D and  H have the same measure (congruent) EXIT TOPICSBACKNEXT

31. Having knowledge of one angle in the special transversal below, allows us to deduce the rest of the angles. l1l1 l2l2 120° C B D FE GH l 1 l 2 What are the measures of the other angles? EXIT TOPICSBACKNEXT

32. Answer: l1l1 l2l2 120°60° l 1 l 2 60° 120° 60° 120° Why? EXIT TOPICSBACKNEXT

33. Triangles EXIT TOPICSBACKNEXT

34. One of the most familiar geometric objects is the triangle. In fact, trigonometry is the study of triangles EXIT TOPICSBACKNEXT

35. Triangles have two important properties 1. 3 sides 2. 3 interior angles A BC EXIT TOPICSBACKNEXT

36. We also have some special triangles. EXIT TOPICSBACKNEXT

37. Right Triangle — One interior angle of the triangle measures 90° (has a right angle) EXIT TOPICSBACKNEXT

38. Equilateral Triangle — 1. All of the sides are congruent (have the same measure). EXIT TOPICSBACKNEXT

39. Equiangular Triangle — 1. All of the interior angles are congruent (have the same measure). EXIT TOPICSBACKNEXT

40. Note – Equiangular triangles are also equilateral triangles. Equilateral triangles are also equiangular triangles. EXIT TOPICSBACKNEXT

41. Isosceles Triangle — 1. Two of the interior angles of the triangle are congruent (have the same measure). 2. Two of the sides are congruent. EXIT TOPICSBACKNEXT

42. The sum of the interior angles of any triangle measures 180° A BC That is,  A +  B +  C = 180° EXIT TOPICSBACKNEXT

43. Why? EXIT TOPICSBACKNEXT

44. Form a transversal with two parallel lines. A BC EXIT TOPICSBACKNEXT

45. Fill in the missing vertical angles. A BC EXIT TOPICSBACKNEXT

46. Solution-- A BC A BC EXIT TOPICSBACKNEXT

47. Fill in the remaining angles. A BC A BC EXIT TOPICSBACKNEXT

48. Solution-- A BC A BC Do you notice anything? BC EXIT TOPICSBACKNEXT

49. That is,  B +  A +  C = 180° A BC A BC Note – The order in which we add doesn’t matter. BC EXIT TOPICSBACKNEXT

50. A BC  A +  B +  C = 180° (This is true for any triangle) EXIT TOPICSBACKNEXT

51. End of Review of Geometry Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT TOPICSBACKNEXT