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Review of Geometry

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

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Review of Geometry

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  1. Review of Geometry Prepared by Title V Staff:Daniel Judge, InstructorKen Saita, Program SpecialistEast Los Angeles College Click one of the buttons below or press the enter key TOPICS BACK NEXT EXIT © 2002 East Los Angeles College. All rights reserved.

  2. Topics Click on the topic that you wish to view . . . LinesAnglesTriangles TOPICS BACK NEXT EXIT

  3. Lines TOPICS BACK NEXT EXIT

  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 2 Region 3 TOPICS BACK NEXT EXIT

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

  6. Parallel Lines – Lines that never intersect. l1 l2 Notationl1 l2 TOPICS BACK NEXT EXIT

  7. Transversal – A line that intersects two or more lines in different points. l1 l2 Note:l1 is not parallel to l2(l1 l2) TOPICS BACK NEXT EXIT

  8. Transversal l1 l2 Note: l1 is parallel to l2(l1 l2) TOPICS BACK NEXT EXIT

  9. Angles TOPICS BACK NEXT EXIT

  10. Angles are formed when lines intersect. l1 A Note: (l1 l2) D B C l2 TOPICS BACK NEXT EXIT

  11. A and B are said to be adjacent. (neighbors) l1 A D B C l2 TOPICS BACK NEXT EXIT

  12. Adjacent Angles – Angles that share a common vertex and a common side between them. l1 A D B C l2 TOPICS BACK NEXT EXIT

  13. l1 A D B C l2 Note:B and C are adjacent (neighbors)C and D are adjacent (neighbors)D and A are adjacent (neighbors) TOPICS BACK NEXT EXIT

  14. Vertical Angles – The pairs of non-adjacent angles formed by the intersection of two lines. l1 A D B C l2 TOPICS BACK NEXT EXIT

  15. l1 A D B C l2 Note:A and C are vertical anglesB and D are vertical angles TOPICS BACK NEXT EXIT

  16. Q: What’s special about vertical angles? Answer – They have the same measure. (they are congruent) l1 110° 70° 70° 110° l2 TOPICS BACK NEXT EXIT

  17. Fact – When you intersect two lines at a point l1 A D B C l2 AC (congruent)B D (congruent) TOPICS BACK NEXT EXIT

  18. Two angles are said to be supplementary if their sum measures 180°.Adjacent angles formed by two intersecting lines are supplementary. l1 A D B C l2 A and B are supplementary angles. TOPICS BACK NEXT EXIT

  19. Can you find any other supplementary angles in the figure below? l1 A D B C l2 TOPICS BACK NEXT EXIT

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

  21. Revisiting the transversal, copy this picture in your notebook. A B l1 C D Note: (l1 l2) E F G l2 H TOPICS BACK NEXT EXIT

  22. Angles in the interior region between the two lines are called interior angles. Angles in the exterior region are called exterior angles. Exterior A B l1 C D Interior Interior E F G l2 H Exterior TOPICS BACK NEXT EXIT

  23. Q: Which are the interior angles and exterior angles? A B l1 C D E F G l2 H TOPICS BACK NEXT EXIT

  24. A B l1 C D E F G l2 H Answer— Interior ExteriorCADBEGFH TOPICS BACK NEXT EXIT

  25. Q: Which angles are adjacent?Q: Which angles are vertical?Q: Which angles are supplementary? A B l1 C D E F G l2 H TOPICS BACK NEXT EXIT

  26. Consider a transversal consisting of the two parallel lines. A B l1 C D E F l2 G H TOPICS BACK NEXT EXIT

  27. A B l1 C D E F l2 G H We know, A  DBCEHGF since they are all vertical angles. TOPICS BACK NEXT EXIT

  28. Q: Are any other angles congruent? TOPICS BACK NEXT EXIT

  29. Yes! If we could slide l2 up to l1, wewould be looking at the following picture. TOPICS BACK NEXT EXIT

  30. A B l1 C D E F l2 G H 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) TOPICS BACK NEXT EXIT

  31. Having knowledge of one angle in the special transversal below, allows us to deduce the rest of the angles. 120° B l1 C D l1 l2 E F l2 G H What are the measures of the other angles? TOPICS BACK NEXT EXIT

  32. Answer: 120° 60° l1 60° 120° l1 l2 120° 60° l2 60° 120° Why? TOPICS BACK NEXT EXIT

  33. Triangles TOPICS BACK NEXT EXIT

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

  35. Triangles have two important properties 1. 3 sides 2. 3 interior angles A B C TOPICS BACK NEXT EXIT

  36. We also have some special triangles. TOPICS BACK NEXT EXIT

  37. Right Triangle —One interior angle ofthe triangle measures90° (has a right angle) TOPICS BACK NEXT EXIT

  38. Equilateral Triangle —1. All of the sides are congruent (have the samemeasure). TOPICS BACK NEXT EXIT

  39. Equiangular Triangle —1. All of the interior anglesare congruent (have the same measure). TOPICS BACK NEXT EXIT

  40. Note – Equiangulartriangles are alsoequilateral triangles. Equilateral triangles are also equiangular triangles. TOPICS BACK NEXT EXIT

  41. Isosceles Triangle —1. Two of the interior angles of the triangle are congruent (havethe same measure).2. Two of the sidesare congruent. TOPICS BACK NEXT EXIT

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

  43. Why? TOPICS BACK NEXT EXIT

  44. Form a transversal with two parallel lines. A B C TOPICS BACK NEXT EXIT

  45. Fill in the missing vertical angles. A B C TOPICS BACK NEXT EXIT

  46. Solution-- A A B C B C TOPICS BACK NEXT EXIT

  47. Fill in the remaining angles. A A B C B C TOPICS BACK NEXT EXIT

  48. Solution-- A B C A B C B C Do you notice anything? TOPICS BACK NEXT EXIT

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

  50. A B C A + B + C = 180°(This is true for any triangle) TOPICS BACK NEXT EXIT

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