1. Slide 9-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION

2. Copyright © 2005 Pearson Education, Inc. Chapter 9 Geometry

3. Copyright © 2005 Pearson Education, Inc. 9.1 Points, Lines, Planes, and Angles

4. Slide 9-4 Copyright © 2005 Pearson Education, Inc. Basic Terms A line segment is part of a line between two points, including the endpoints. Line segment AB Ray BA Ray AB Line AB SymbolDiagramDescription A B A A A B B B

5. Slide 9-5 Copyright © 2005 Pearson Education, Inc. Plane Any three points that are not on the same line (noncollinear points) determine a unique plane. A line in a plane divides the plane into three parts, the line and two half planes. Any line and a point not on the line determine a unique plane. The intersection of two planes is a line.

6. Slide 9-6 Copyright © 2005 Pearson Education, Inc. Angles The measure of an angle is the amount of rotation from its initial to its terminal side. Angles can be measured in degrees, radians, or, gradients. Angles are classified by their degree measurement.  Right Angle is 90   Acute Angle is less than 90   Obtuse Angles is greater than 90  but less than 180   Straight Angle is 180 

7. Slide 9-7 Copyright © 2005 Pearson Education, Inc. Types of Angles Adjacent Angles-angles that have a common vertex and a common side but no common interior points. Complementary Angles-two angles whose sum is 90 degrees. Supplementary Angles-two angles whose sum is 180 degrees.

8. Slide 9-8 Copyright © 2005 Pearson Education, Inc. Example If are supplementary and the measure of ABC is 6 times larger than CBD, determine the measure of each angle. A B C D

9. Slide 9-9 Copyright © 2005 Pearson Education, Inc. More definitions Vertical angles have the same measure. A line that intersects two different lines, at two different points is called a transversal. Special angles are given to the angles formed by a transversal crossing two parallel lines.

10. Slide 9-10 Copyright © 2005 Pearson Education, Inc. Special Names 56 12 4 87 3 One interior and one exterior angles on the same side of the transversal-have the same measure Corresponding angles Exterior angles on the opposite sides of the transversal—have the same measure Alternate exterior angles Interior angles on the opposite side of the transversal—have the same measure Alternate interior angles 56 12 4 87 3 56 12 4 87 3

11. Copyright © 2005 Pearson Education, Inc. 9.2 Polygons

12. Slide 9-12 Copyright © 2005 Pearson Education, Inc. Polygons Polygons are names according to their number of sides. Icosagon20Heptagon7 Dodecagon12Hexagon6 Decagon10Pentagon5 Nonagon9Quadrilateral4 Octagon8Triangle3 NameNumber of Sides NameNumber of Sides

13. Slide 9-13 Copyright © 2005 Pearson Education, Inc. Triangles The sum of the measures of the interior angles of an n-sided polygon is (n  2)180 . Example: A certain brick paver is in the shape of a regular octagon. Determine the measure of an interior angle and the measure of one exterior angle.

14. Slide 9-14 Copyright © 2005 Pearson Education, Inc. Triangles continued Determine the sum of the interior angles. The measure of one interior angle is The exterior angle is supplementary to the interior angle, so 180   135  = 45 

15. Slide 9-15 Copyright © 2005 Pearson Education, Inc. Types of Triangles

16. Slide 9-16 Copyright © 2005 Pearson Education, Inc. Similar Figures Two polygons are similar if their corresponding angles have the same measure and their corresponding sides are in proportion. 4 3 4 6 66 9 4.5

17. Slide 9-17 Copyright © 2005 Pearson Education, Inc. Example Catherine Johnson wants to measure the height of a lighthouse. Catherine is 5 feet tall and determines that when her shadow is 12 feet long, the shadow of the lighthouse is 75 feet long. How tall is the lighthouse? x 75 12 5

18. Slide 9-18 Copyright © 2005 Pearson Education, Inc. Example continued x 75 12 5 Therefore, the lighthouse is 31.25 feet tall.

19. Slide 9-19 Copyright © 2005 Pearson Education, Inc. Congruent Figures If corresponding sides of two similar figures are the same length, the figures are congruent. Corresponding angles of congruent figures have the same measure.

20. Slide 9-20 Copyright © 2005 Pearson Education, Inc. Quadrilaterals Quadrilaterals are four-sided polygons, the sum of whose interior angles is 360 . Quadrilaterals may be classified according to their characteristics.

21. Slide 9-21 Copyright © 2005 Pearson Education, Inc. Classifications Trapezoid Two sides are parallel. Parallelogram Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length.

22. Slide 9-22 Copyright © 2005 Pearson Education, Inc. Classifications continued Rhombus Both pairs of opposite sides are parallel. The four sides are equal in length. Rectangle Both pairs of opposite sides are parallel. Both pairs of opposite sides are equal in length. The angles are right angles.

23. Slide 9-23 Copyright © 2005 Pearson Education, Inc. Classifications continued Square Both pairs of opposite sides are parallel. The four sides are equal in length. The angles are right angles.

24. Copyright © 2005 Pearson Education, Inc. 9.3 Perimeter and Area

25. Slide 9-25 Copyright © 2005 Pearson Education, Inc. Formulas P = s 1 + s 2 + b 1 + b 2 P = s 1 + s 2 + s 3 P = 2b + 2w P = 4s P = 2l + 2w Perimeter Trapezoid Triangle A = bhParallelogram A = s 2 Square A = lwRectangle AreaFigure

26. Slide 9-26 Copyright © 2005 Pearson Education, Inc. Example Marcus Sanderson needs to put a new roof on his barn. One square of roofing covers 100 ft 2 and costs $32.00 per square. If one side of the barn roof measures 50 feet by 30 feet, determine a) the area of the entire roof. b) how many squares of roofing he needs. c) the cost of putting on the roof.

27. Slide 9-27 Copyright © 2005 Pearson Education, Inc. Example continued a) The area of the roof is  A = lw A = 30(50) A = 1500 ft 2 1500(2 both sides of the roof) = 3000 ft 2 b) Determine the number of squares

28. Slide 9-28 Copyright © 2005 Pearson Education, Inc. Example continued c) Determine the cost  30 squares  $32 per square  $960  It will cost a total of $960 to roof the barn.

29. Slide 9-29 Copyright © 2005 Pearson Education, Inc. Pythagorean Theorem

30. Slide 9-30 Copyright © 2005 Pearson Education, Inc. Example Tomas is bringing his boat into a dock that is 12 feet above the water level. If a 38 foot rope is attached to the dock on one side and to the boat on the other side, determine the horizontal distance from the dock to the boat. 12 ft 38 ft rope

31. Slide 9-31 Copyright © 2005 Pearson Education, Inc. Example continued The distance is approximately 36.06 feet.

32. Slide 9-32 Copyright © 2005 Pearson Education, Inc. Circles A circle is a set of points equidistant from a fixed point called the center. A radius, r, of a circle is a line segment from the center of the circle to any point on the circle. A diameter, d, of a circle is a line segment through the center of the circle with both end points on the circle.

33. Slide 9-33 Copyright © 2005 Pearson Education, Inc. Example Terri is installing a new circular swimming pool in her backyard. The pool has a diameter of 27 feet. How much area will the pool take up in her yard? The radius of the pool is 13.5 feet. The pool will take up about 573 square feet.

34. Copyright © 2005 Pearson Education, Inc. 9.4 Volume

35. Slide 9-35 Copyright © 2005 Pearson Education, Inc. Volume Volume is the measure of the capacity of a figure.

36. Slide 9-36 Copyright © 2005 Pearson Education, Inc. Formulas Sphere Cone V =  r 2 h Cylinder V = s 3 Cube V = lwhRectangular Solid DiagramFormulaFigure

37. Slide 9-37 Copyright © 2005 Pearson Education, Inc. Example Mr. Stoller needs to order potting soil for his horticulture class. The class is going to plant seeds in rectangular planters that are 12 inches long, 8 inches wide and 3 inches deep. If the class is going to fill 500 planters, how many cubic inches of soil are needed?

38. Slide 9-38 Copyright © 2005 Pearson Education, Inc. Example continued We need to find the volume of one planter. Soil for 500 planters would be  500(288) = 144,000 cubic inches 

39. Slide 9-39 Copyright © 2005 Pearson Education, Inc. Polyhedron A polyhedron is a closed surface formed by the union of polygonal regions.

40. Slide 9-40 Copyright © 2005 Pearson Education, Inc. Euler’s Polyhedron Formula Number of vertices  number of edges + number of faces = 2 Example: A certain polyhedron has 12 edges and 6 faces. Determine the number of vertices on this polyhedron. Number of vertices  number of edges + number of faces = 2 There are 8 vertices.

41. Slide 9-41 Copyright © 2005 Pearson Education, Inc. Volume of a Prism V = Bh, where B is the area of the base and h is the height. Example: Find the volume of the figure.  Area of one triangle. Find the volume. 8 m 6 m 4 m

42. Slide 9-42 Copyright © 2005 Pearson Education, Inc. Volume of a Pyramid where B is the area of the base and h is the height. Example: Find the volume of the pyramid.  Base area = 12 2 = 144  12 m 18 m

43. Copyright © 2005 Pearson Education, Inc. 9.5 Transformational Geometry, Symmetry, and Tessellations

44. Slide 9-44 Copyright © 2005 Pearson Education, Inc. Definitions The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation).

45. Slide 9-45 Copyright © 2005 Pearson Education, Inc. Reflection A reflection is a rigid motion that moves a a geometric figure to a new position such that the figure in the new position is a mirror image of the figure starting position. In two dimensions the figure and its mirror image are equidistant from a line called the reflection line or the axis of reflection.

46. Slide 9-46 Copyright © 2005 Pearson Education, Inc. Construct the reflection of triangle ABC about the line l. A B C ll A B C B’ A’ C’ 2 units

47. Slide 9-47 Copyright © 2005 Pearson Education, Inc. Translation A translation (or glide) is a rigid motion that moves a geometric figure by sliding it along a straight line segment in the plane. The direction and length of the line segment completely determine the translation.

48. Slide 9-48 Copyright © 2005 Pearson Education, Inc. Example

49. Slide 9-49 Copyright © 2005 Pearson Education, Inc. Example continued

50. Slide 9-50 Copyright © 2005 Pearson Education, Inc. Rotation A rotation is a rigid motion performed by rotating a geometric figure in the plane about a specific point, called the rotation point or the center of rotation. The angle through which the object is rotated is called the angle of rotation.

51. Slide 9-51 Copyright © 2005 Pearson Education, Inc. Example

52. Slide 9-52 Copyright © 2005 Pearson Education, Inc. Example continued

53. Slide 9-53 Copyright © 2005 Pearson Education, Inc. Glide Reflection A glide reflection is a rigid motion formed by performing a translation (or glide) followed by a reflection.

54. Slide 9-54 Copyright © 2005 Pearson Education, Inc. Example

55. Slide 9-55 Copyright © 2005 Pearson Education, Inc. Symmetry A symmetry of a geometric figure is a rigid motion that moves a figure back onto itself. That is, the beginning position and ending position of the figure must be identical.

56. Slide 9-56 Copyright © 2005 Pearson Education, Inc. Example

57. Slide 9-57 Copyright © 2005 Pearson Education, Inc. Tessellations A tessellation (or tiling) is a pattern of the repeated use of the same geometric figures to entirely cover a plane, leaving no gaps. The geometric figures use are called the tessellating shapes of the tessellation.

58. Slide 9-58 Copyright © 2005 Pearson Education, Inc. Example The simplest tessellations use one single regular polygon.

59. Slide 9-59 Copyright © 2005 Pearson Education, Inc. Example continued