Chapter 8 Geometry

8.1 Lines and Angles

Identifying Lines, Line Segments. Rays, and Angles Space extends in all directions indefinitely. A Plane is a flat surface that extends indefinitely. Objective A

Identifying Lines, Line Segments. Rays, and Angles A point has no length, no width, and no height, but it does have a location. P Point P A line is set of points extending indefinitely in two directions. Objective A A B Line AB or AB 4

Identifying Lines, Line Segments. Rays, and Angles Definition Example Line Segment - A piece of a line with two endpoints. Ray - A part of a line with one endpoint. Angle - Made up of two rays that share the same endpoint. Vertex - The common endpoint of an angle. A B Line segment AB or A B Ray AB or A B C Vertex Objective A 5

Example Identify each figure as a line, a ray, a line segment, or an angle. Then name the figure using the given point. a. b. c. This is a ray. It is ray FG or F G This is a line segment. It is line segment NP or N P Objective A • This is a line. It is line RS or R S 6

Classifying Angles as Acute, Right, Obtuse, or Straight Definition Example Degrees - used to measure angles, symbolized by a small raised circle . Straight angle - an angle that measures 180. Right angle - an angle that measures 90. Objective A 7

Classifying Angles as Acute, Right, Obtuse, or Straight Definition Example Acute Angle - an angle whose measure is between 0 and 90. Obtuse angle - an angle that measures between 90 and 180. Objective A 8

Identifying Complementary and Supplementary Angles Definition Example Complementary Angles – two angles that have a sum of 90. Supplementary Angles – two angles that have a sum of 180. Objective A 9

Example Find the complement of a 62 angle. Two angles that have a sum of 90 are complementary. Objective A 10

Example Find the supplement of a 121 angle. Two angles that have a sum of 180 are supplementary. Objective A 11

Finding Measures of Angles Two lines in a place can either be parallel or intersection. Parallel lines never meet. The symbol‖is used to indicate “parallel to”. Perpendicular lines form right angles when they intersect. The symbol is used to denote perpendicular lines. Objective A 12

Vertical and Adjacent Angles When two lines intersect, four angles are formed. Two angles that are opposite each other are called vertical angles. Vertical angles have the same measure. Two angles that share a common side are called adjacent angles. Adjacent angles formed by intersecting lines are supplementary. Objective A 13

Parallel Lines Parallel Line Cut by a Transversal If two parallel lines are cut by a transversal, then the measure of corresponding angles are equal and the measure of the alternate interior angles are equal. Objective A 14

Example m || n, find the measures of x, y, and z. 80° z x y Objective A 15

Plane Figures and Solids 8.2 Plane Figures and Solids

Identifying Plane Figures A plane figure is a figure that lies on a plane. A plane figure, like planes, have length and width but no thickness or depth. Objective A

Polygons Definition Example Polygon - A closed plane figure that basically consists of three or more line segments that meet their end points. Regular polygon - A closed plane figure whose sides are all the same length and whose angles are the same measure. A B C D E F G H I J E Objective A 18

Polygons Polygons Number of Sides Example Figure Examples 3 Triangle A, F 4 Quadrilateral B,E,G 5 Pentagon H 6 Hexagon I 7 Heptagon C 8 Octagon J 9 Nonagon K 10 Decagon D Objective A Refer to the figures on the previous slide. 19

Example Find the measure of Since the sum of the measures of the three angles is 180̊ , we have measure of Objective A 20

Triangle Classification Triangles Triangle Classification Equilateral Isosceles Scalene All three sides are the same length. Also, all three angles have the same measure. Two sides are the same length. Also, the angles opposite the equal sides have equal measure. No sides are the same length. No angles have the same measure. Objective A 21

Example Find the measure of We know that the measure of the right angle is 90. Since the sum of the measures of the angles is 180, we have Objective A 22

Special Quadrilaterals Parallelogram – opposite sides are parallel and equal in length. Rectangle – special parallelogram that has four right angles. Square – special rectangle that has all four sides equal in length. Rhombus – special parallelogram that has all four sides equal in length. Objective A Trapezoid – quadrilateral with exactly one pair of opposite sides parallel. 23

Circles Definition Example Circle - A plane figure that consists of all points that are the same fixed distance from a point c, called the center. Radius - The distance from the center of the circle to any point on the circle. Diameter - The distance across the circle passing through the center. center radius diameter Objective A 24

Identifying Solid Figures Definition Example Solid - A figure that lies in space and has length, width, and height or depth. Rectangular solid - A solid that consists of six sides, or faces, all of which are rectangles. Cube - A rectangular solid whose six sides are squares. Pyramid - The pyramids we will study have square bases and heights that are perpendicular to their base. Width Height Length Objective A Height Square base 25

Spheres Definition Example Sphere - Consists of all points in space that are the same distance from a point c, called the center of the sphere Radius - The distance from the center to the sphere. Diameter - The distance across the sphere passing through the center. Radius Diameter Center Objective A 26

Example Find the radius of the sphere. The radius is half the diameter. Objective A The radius is 18 feet. 27

8.3 Perimeter

Example Find the perimeter of the rectangle. length width 4 m 10 m Objective A 29

Perimeter of a Rectangle length width width length Objective A In symbols, this can be written as P = 2 · l + 2 · w 30

Example Find the perimeter of the rectangle with a length of 8 inches and a width of 5 inches. 5 in. 8 in. P = 2 · l + 2 · w = 2 · 8 in. + 2 · 5 in. Objective A = 16 in. + 10 in. = 26 in. The perimeter is 26 inches. 31

Perimeter of a Square Perimeter of a Square In symbols, P = 4 · s side Objective A 32

Perimeter of a Triangle In symbols, P = a + b + c side b side a side c Objective A 33

Example Find the perimeter of the triangle if the sides are 4 meters, 7 meters and 9 meters. P = a + b + c 4 meters 10 meters 9 meters P = 4 + 9 + 10 Objective A = 23 m The perimeter is 23 meters. 34

Example Find the perimeter of the following figure. 30 in. – 13 in. = 17 in. 6 in. 13 in. 30 in. Objective A 13 in. – 6 in. = 7 in. P = 13 + 13 + 30 + 7 + 17 + 6 P = 86 The perimeter is 86 inches 35

Using Formulas to Find Circumference Circumference of a Circle center radius diameter Objective A 36

Example Find the circumference of the circle. 10 yd Objective A The circumference is about 62.8 yards. 37