This document provides an overview of key concepts in plane geometry covered in Chapter 5, including points, lines, planes, angles, parallel and perpendicular lines, triangles, polygons, coordinate geometry, congruence, transformations, and tessellations. Specific topics discussed include classifying angles, properties of parallel lines cut by a transversal, the triangle sum theorem, types of triangles and polygons, finding total angle measures of polygons, and using coordinate geometry. Homework problems are provided at the end of sections for additional practice.

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. 5-1 Points, Lines, Planes & Angles Vocabulary Point – Names a location Line – Perfectly straight and extends in both directions forever Plane - Perfectly flat surface that extends forever in all directions Segment – Part of a line between two points Ray – Part of a line that starts at a point and extends forever in one direction

3. Point

4. Line

5. Segment

6. Ray

7. Example 1 Name four points Name the line Name the plane Name four segments Name five rays

8. More Vocabulary Right Angle – Measures exactly 90 ° Acute Angle – Measures less than 90 ° Obtuse Angle – Measures more than 90 ° Complementary Angle – Angles that measure 90 ° together Supplementary Angle – Angles that measure 180 ° together

9. Right Angle

10. Acute Angle

11. Obtuse Angle

12. Complementary Angle

13. Supplementary Angle

14. Example 2 Name the following: Right Angle Acute Angle Obtuse Angle Complementary Angle Supplementary Angle

15. Even MORE Vocabulary Congruent – Figures that have the same size AND shape Vertical Angles Angles A & C are VA Angles B & D are VA If Angle A is 60 ° what is the measure of angle B?

16. Homework/Classwork Page 225, #13-34

17. 5-2 Parallel and Perpendicular Lines Vocabulary Parallel Lines – Two lines in a plane that never meet, ex. Railroad Tracks Perpendicular Lines – Lines that intersect to form Right Angles Transversal – A line that intersects two or more lines at an angle other than a Right Angle

18. Parallel Lines

19. Perpendicular Lines

20. Transversal

21. Transversals to parallel lines have interesting properties The color coded numbers are congruent

22. Properties of Transversals to Parallel Lines If two parallel lines are intersected by a transversal: The acute angles formed are all congruent The obtuse angles are all congruent And any acute angle is supplementary to any obtuse angle If the transversal is perpendicular to the parallel lines, all of the angles formed are congruent 90° angles

23. Alternate Interior Angles

24. Alternate Exterior Angles

25. Corresponding Angles

26. Symbols Parallel Perpendicular Congruent

27. Example 1 In the figure Line X Y Find each angle measure

29. In the figure Line A B Find each angle measure

31. Homework/Classwork Page 230, # 6-20

32. 5-3 Triangles Triangle Sum Theorem – The angle measures of a triangle in a plane add to 180° Because of alternate interior angles, the following is true:

33. Vocabulary Acute Triangle – All angles are less than 90° Right Triangle – Has one 90° angle Obtuse Triangle – Has one obtuse angle

34. Example Find the missing angle

35. Example Find the missing angle.

36. Example Find the missing angles

37. Vocabulary Equilateral Triangle – 3 congruent sides and angles Isosceles Triangle – 2 congruent sides and angles Scalene Triangle – No congruent sides or angles

38. Equilateral Triangle Isosceles Triangle Scalene Triangle

39. Remember…they are ALL triangles

40. Example Find the missing angle(s)

41. Example Find the missing angle(s)

42. Example Find the missing angle(s)

43. Example Find the angles. Hint, remember the triangle sum theorem

45. Classwork/Homework Page 237, #10-26

46. 5-4 Polygons Polygons Have 3 or more sides Named by the number of sides “ Regular Polygon” means that all the sides are equal length n n-gon 8 Octagon 7 Heptagon 6 Hexagon 5 Pentagon 4 Quadrilateral 3 Triangle # of Sides Polygon

47. Finding the sum of angles in a polygon Step 1: Divide the polygon into triangles with common vertex

48. Step 2: Multiply the number of triangles by 180

49. The Short Cut 180°( n – 2) where n = the number of angles in the figure In this case n = 6 = 180°(6 – 2) = 180°(4) = 720° *Notice that n - 2 = 4 **Also notice that the figure can be broken into 4 triangles…coincidence? I don’t think so!

51. Example Find the missing angle

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

53. Classwork/Homework Page 242, # 13-24