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Section 9.1 Geometry Introduction
Math 310 Section 9.1 Geometry Introduction
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Axiomatic System A logical system which possesses an explicitly stated set of axioms from which theorems can be derived. (mathworld.wolfram.com)
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Undefined Terms Point Line Plane
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“One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs.” - David Hilbert
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Linear “Notions” Collinear points Is between Line segment (segment)
Ray
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Collinear points (& non collinear)
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Is Between
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Line Segment
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Ray
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Planar “Notions” Coplanar points Noncoplanar points Coplanar lines
Skew lines Intersecting lines Concurrent lines Parallel lines
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Coplanar & Noncoplanar Points
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Coplanar lines
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Skew Lines
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Intersecting Lines
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Concurrent Lines
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Parallel Lines
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Properties of “tables, chairs and beer mugs”
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.
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Properties (cont) A line and a point not on the line determine a plane. Two parallel lines determine a plane Two intersecting lines determine a plane.
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Property 1 A B A B
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Property 2
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Property 3
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Property 4
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Property 5
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Property 6
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Property 7
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Intersecting Planes Parallel Along a line
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Parallel Planes
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Planes Intersecting along a line
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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.
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Ex. Angle: <ABC Vertex Side
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Ex. <EBF <GFE <I
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Ex. <MJK Name all six angles. <NKL <OLJ <KJL <LKJ
<JLK Name all six angles.
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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.
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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”
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Ex. Solve for x. m<ABC = 80° m<ABD = 30° m<DBC = (x – 25)°
m<ABD = (x – 13)° m<DBC = (x + 7)° x = 44
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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.
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Types of Angles Obtuse Acute Right Straight
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Ex. acute obtuse straight right
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Perpendicular Lines Def.
Two lines are perpendicular if they intersect and form right angles. perpendicular not perpendicular
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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.
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Ex.
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Ex.
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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
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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.
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