Transcript Section 8.1
Chapter 8 Introductory Geometry Section 8.1 Beginning Geometry Geometry Geometry is the study of shapes and is a part of the early childhood, middle childhood and adolescent young adult mathematics curriculum. Geometry analyzes patterns in shapes using both numbers and logical reasoning. Two-Dimensional vs Three-Dimensional Figures A plane is a flat sheet. Think of it like a very thin piece of paper that does not bend. A plane figure is some shape that is part of or inside a plane. Plane figures are considered to be two-dimensional shapes or flat. Three-dimensional shapes are shapes that have length, height and depth to them. Two-Dimensional Shapes Three-Dimensional Shapes Points, Lines, Segments, Rays and Angles The geometric objects points, lines, segments, rays and angle are frequently occurring components of other more complicated geometric objects. The description of a these objects has both a formal definition which is used in the later grades and an intuitive everyday situation model. A point represents a location in a plane, space or on an object. Cities on a map are an everyday situation of how points are represented. Points are thought of as dots and usually labeled with capital letters. Points A, B and C are shown below the map. A D A line is straight and extends infinitely in both directions. Two points on a line can name it. A line segment is part of a line between two points. The two points are called the endpoints and name the segment. A ray is the part of a line that extends infinitely in one direction. Two points on a ray can name it with one point being the end of the ray. DE E F H C B G I FG HI Angles Angles are made up of the union of two rays that have a common endpoint. The two rays are called the sides of the angle and the common endpoint is called the vertex of the angle. Using the set operations of intersection () and union () to describe geometric shapes is very common. Angles are named with three points with the point in the middle being the vertex and the other two being a point on each side and we use the symbol to mark an angle. BAC AB AC A F C B E D BAC DAE AD AE AC CA AC Angles are measured in a unit called degrees with a device named a protractor. The vertex is placed at the center and one side at the zero degree mark. The place where the other side crosses is the measure. The measure is denoted with m. 120 90 60 B 30 150 0 180 A mBAC 60 C Compass directions are often given using degrees. The picture to the right show how to locate a plane whose position is described a 30 south of east. east 30 south Types of Angles Adjectives are used to describe certain types of angles. It can refer to its size, or the relationship it has with another angle. Many of these have both a numerical and geometric way of characterizing them. D Right Angles These are angles that measure 90. They break a line into two angles that are exactly the same. DAB is exactly the same as DAC C A B mDAB 90 mDAC Straight Angles These are angles that measure 180. They form a line. To the right BAC is a straight angle. C A B mBAC 180 Acute Angles Obtuse Angles These are angles whose measure is less than 90. C E 38 A B D F Reflex Angles H 326 G These are the angles whose measure exceeds 180. They make up the bigger part of the angle. I Complementary Angles Supplementary Angles These are pairs of angles that make up each part of a right angle. The measures will add up to 90. These are pairs of angles that make up each part of a straight angle. The measures will add up to 180. B BAC and CAD are complementary angles. C 68 22 A These are angles whose measure is greater than 90 and less than 180. 147 D C 158 22 B A D BAC and CAD are supplementary angles. Congruent Figures Congruent geometric figures are figures that can be slid one on top of another so that they will exactly match up. Congruent angles are angles that can be placed one on top of the other so that they coincide. B E BAC is congruent to EDF A D C F Intersecting Lines Lines that intersect have exactly one point in common. Lines (in the same plane) that do not intersect are called parallel and are denoted with ||. Lines that intersect at a right angle are called perpendicular and are denoted with . In the figure to the right we have: AB || CD AB AD B A C D