Transcript 9-1
Math 310 Section 9.1 Geometry Introduction Axiomatic System A logical system which possesses an explicitly stated set of axioms from which theorems can be derived. (mathworld.wolfram.com) www.xkcd.com Undefined Terms Point Line Plane “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” - David Hilbert Linear “Notions” Collinear points Is between Line segment (segment) Ray Collinear points (& non collinear) D B E C Is Between B E C Line Segment F G Ray I H Planar “Notions” Coplanar points Noncoplanar points Coplanar lines Skew lines Intersecting lines Concurrent lines Parallel lines Coplanar & Noncoplanar Points D B A C Coplanar lines I E D F G H Skew Lines I E F H Intersecting Lines J L K Concurrent Lines J M K L Parallel Lines O N P Q Properties of “tables, chairs and beer mugs” 1. 2. 3. 4. There is exactly one line that contains any two distinct points If two points lie in a plane, then the line containing the points lies in the plane. If two distinct planes intersect, then their intersection is a line. There is exactly one plane that contains any three distinct noncollinear points. Properties (cont) 5. 6. 7. A line and a point not on the line determine a plane. Two parallel lines determine a plane Two intersecting lines determine a plane. Property 1 B A A B Property 2 Property 3 B A Property 4 Property 5 Property 6 Property 7 Intersecting Planes Parallel Along a line Parallel Planes Planes Intersecting along a line Angle, Vertex, Side Def When two rays share an endpoint, an angle is formed. The common initial point of the rays is the vertex of the angle. Each ray is called a side of the angle. Ex. A B Angle: <ABC Side Vertex C Ex. E H E G F <EBF B I F J <I <GFE Ex. M J K N <MJK Name all six angles. <NKL <OLJ L O <KJL <LKJ <JLK Angle Measure To measure an angle we use the unit degree. It measures the “opening” of the angle. The largest angle measure is 360° and the smallest is 0°. A complete rotation about a point is 360°. For more accuracy, angles can be further measure in minutes, and seconds. Each degree is divided into 60 minutes, and each minute is divided into 60 seconds. Ex. Add: 45°23’47” and 62°36’51” 45°23’47” + 62°36’51” = 108°0’38” or 108°38” Add: 145°17’4” and 220°31’32” 145°17’4” + 220°31’32” = 365°48’36” = 5°48’36” Ex. Solve for x. m<ABC = 80° m<ABD = 30° m<DBC = (x – 25)° x = 75 A C m<ABC = 82° m<ABD = (x – 13)° m<DBC = (x + 7)° x = 44 D B Protractor A protractor is a standard tool for measuring angles. To use, line the vertex up with the center of the base of the protractor and line one side of the angle up with the 0° mark. Now measure from 0°, increasing, until you see the other side of the angle and read the mark. Types of Angles Obtuse Acute Right Straight E Ex. H B F acute I H G C B J obtuse F straight A right Perpendicular Lines Def. Two lines are perpendicular if they intersect and form right angles. perpendicular not perpendicular Line Perpendicular to a Plane A line is perpendicular to a plane if it is perpendicular to every line, contained in the plane, passing through the point of intersection. Ex. Ex. Questions Is it possible for a line intersecting a plane to be perpendicular to exactly one line in the plane through its intersection with the plane? Can a line intersecting a plane be perpendicular to exactly two distinct lines in the plane going through the point of intersection? Yes No Line Perpendicular to a Plane Thrm A line perpendicular to two distinct lines in the plane through its intersection with the plane is perpendicular to the plane.