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# Plane Geometry

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### Plane Geometry

1. 1. Chapter 5 – Plane Geometry 5-1 Points, Lines, Planes, and Angles 5-2 Parallel and Perpendicular Lines 5-3 Triangles 5-4 Polygons 5-5 Coordinate Geometry 5-6 Congruence 5-7 Transformations 5-8 Symmetry 5-9 Tessellations
2. 2. 5-1 Points, Lines, Planes & Angles <ul><li>Vocabulary </li></ul><ul><li>Point – Names a location </li></ul><ul><li>Line – Perfectly straight and extends in both directions forever </li></ul><ul><li>Plane - Perfectly flat surface that extends forever in all directions </li></ul><ul><li>Segment – Part of a line between two points </li></ul><ul><li>Ray – Part of a line that starts at a point and extends forever in one direction </li></ul>
3. 3. Point
4. 4. Line
5. 5. Segment
6. 6. Ray
7. 7. Example 1 <ul><li>Name four points </li></ul><ul><li>Name the line </li></ul><ul><li>Name the plane </li></ul><ul><li>Name four segments </li></ul><ul><li>Name five rays </li></ul>
8. 8. More Vocabulary <ul><li>Right Angle – Measures exactly 90 ° </li></ul><ul><li>Acute Angle – Measures less than 90 ° </li></ul><ul><li>Obtuse Angle – Measures more than 90 ° </li></ul><ul><li>Complementary Angle – Angles that measure 90 ° together </li></ul><ul><li>Supplementary Angle – Angles that measure 180 ° together </li></ul>
9. 9. Right Angle
10. 10. Acute Angle
11. 11. Obtuse Angle
12. 12. Complementary Angle
13. 13. Supplementary Angle
14. 14. Example 2 <ul><li>Name the following: </li></ul><ul><li>Right Angle </li></ul><ul><li>Acute Angle </li></ul><ul><li>Obtuse Angle </li></ul><ul><li>Complementary Angle </li></ul><ul><li>Supplementary Angle </li></ul>
15. 15. <ul><li>Even MORE Vocabulary </li></ul><ul><li>Congruent – Figures that have the same size AND shape </li></ul><ul><li>Vertical Angles </li></ul><ul><ul><li>Angles A & C are VA </li></ul></ul><ul><ul><li>Angles B & D are VA </li></ul></ul><ul><li>If Angle A is 60 ° what is the measure of angle B? </li></ul>
16. 16. Homework/Classwork <ul><li>Page 225, #13-34 </li></ul>
17. 17. 5-2 Parallel and Perpendicular Lines <ul><li>Vocabulary </li></ul><ul><li>Parallel Lines – Two lines in a plane that never meet, ex. Railroad Tracks </li></ul><ul><li>Perpendicular Lines – Lines that intersect to form Right Angles </li></ul><ul><li>Transversal – A line that intersects two or more lines at an angle other than a Right Angle </li></ul>
18. 18. Parallel Lines
19. 19. Perpendicular Lines
20. 20. Transversal
21. 21. <ul><li>Transversals to parallel lines have interesting properties </li></ul><ul><li>The color coded numbers are congruent </li></ul>
22. 22. Properties of Transversals to Parallel Lines <ul><li>If two parallel lines are intersected by a transversal: </li></ul><ul><ul><li>The acute angles formed are all congruent </li></ul></ul><ul><ul><li>The obtuse angles are all congruent </li></ul></ul><ul><ul><li>And any acute angle is supplementary to any obtuse angle </li></ul></ul><ul><li>If the transversal is perpendicular to the parallel lines, all of the angles formed are congruent 90° angles </li></ul>
23. 23. Alternate Interior Angles
24. 24. Alternate Exterior Angles
25. 25. Corresponding Angles
26. 26. Symbols <ul><li>Parallel </li></ul><ul><li>Perpendicular </li></ul><ul><li>Congruent </li></ul>
27. 27. Example 1 <ul><li>In the figure Line X Y </li></ul><ul><li>Find each angle measure </li></ul>
28. 29. <ul><li>In the figure Line A B </li></ul><ul><li>Find each angle measure </li></ul>
29. 31. Homework/Classwork <ul><li>Page 230, # 6-20 </li></ul>
30. 32. 5-3 Triangles <ul><li>Triangle Sum Theorem – The angle measures of a triangle in a plane add to 180° </li></ul><ul><ul><li>Because of alternate interior angles, the following is true: </li></ul></ul>
31. 33. Vocabulary <ul><li>Acute Triangle – All angles are less than 90° </li></ul><ul><li>Right Triangle – Has one 90° angle </li></ul><ul><li>Obtuse Triangle – Has one obtuse angle </li></ul>
32. 34. Example <ul><li>Find the missing angle </li></ul>
33. 35. Example <ul><li>Find the missing angle. </li></ul>
34. 36. Example <ul><li>Find the missing angles </li></ul>
35. 37. Vocabulary <ul><li>Equilateral Triangle – 3 congruent sides and angles </li></ul><ul><li>Isosceles Triangle – 2 congruent sides and angles </li></ul><ul><li>Scalene Triangle – No congruent sides or angles </li></ul>
36. 38. <ul><li>Equilateral Triangle </li></ul><ul><li>Isosceles Triangle </li></ul><ul><li>Scalene Triangle </li></ul>
37. 39. Remember…they are ALL triangles
38. 40. Example <ul><li>Find the missing angle(s) </li></ul>
39. 41. Example <ul><li>Find the missing angle(s) </li></ul>
40. 42. Example <ul><li>Find the missing angle(s) </li></ul>
41. 43. Example <ul><li>Find the angles. Hint, remember the triangle sum theorem </li></ul>
42. 45. Classwork/Homework <ul><li>Page 237, #10-26 </li></ul>
43. 46. 5-4 Polygons <ul><li>Polygons </li></ul><ul><ul><li>Have 3 or more sides </li></ul></ul><ul><ul><li>Named by the number of sides </li></ul></ul><ul><ul><li>“ Regular Polygon” means that all the sides are equal length </li></ul></ul>n n-gon 8 Octagon 7 Heptagon 6 Hexagon 5 Pentagon 4 Quadrilateral 3 Triangle # of Sides Polygon
44. 47. Finding the sum of angles in a polygon <ul><li>Step 1: </li></ul><ul><ul><li>Divide the polygon into triangles with common vertex </li></ul></ul>
45. 48. <ul><li>Step 2: </li></ul><ul><ul><li>Multiply the number of triangles by 180 </li></ul></ul>
46. 49. The Short Cut <ul><li>180°( n – 2) where n = the number of angles in the figure </li></ul><ul><li>In this case n = 6 </li></ul><ul><li>= 180°(6 – 2) </li></ul><ul><li>= 180°(4) </li></ul><ul><li>= 720° </li></ul>*Notice that n - 2 = 4 **Also notice that the figure can be broken into 4 triangles…coincidence? I don’t think so!
47. 51. Example <ul><li>Find the missing angle </li></ul>
48. 52. This chart may help… n 8 7 6 5 4 3 # of Sides n ° n-gon 1080 ° Octagon 900 ° Heptagon 720 ° Hexagon 540 ° Pentagon 360 ° Quadrilateral 180 ° Triangle Total Angle measure Polygon
49. 53. Classwork/Homework <ul><li>Page 242, # 13-24 </li></ul>