Chapter 8: Informal Geometry-Answers
Section 8.1: An Introduction to Geometry
Points and Lines
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Points: no length, width, or height
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Space: set of all points
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Line: a particular set of points in space; has length but no width or height
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Plane: another set of points in space; has length and width, but no height
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Collinear: points on the same line
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Pass through: line containing a certain set of points is said to pass through the points
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Between: if 3 points are collinear then one of the points is between the other two
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Contain: if point A is on line l, then line l is said to contain point A
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Element: If l contains point A, then A is an element of l
Properties of Lines:
1. If two distinct lines touch or cross (intersect), then the intersection is exactly one point.
2. Two or more lines are said to be concurrent if there is exactly one point common to all of them. (These
lines may be in the same plane, or they may be lines in space.)
3. Exactly one line contains any two distinct points.
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Parallel Lines: Two lines in a plane are parallel if and only if there is no point common to both lines.
Notation for parallel lines: m || l read m is parallel to l
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Skew lines: two lines that do not intersect that are in different planes.
Subset of Lines
1. A half line is a subset of a line, containing all points of the line on one side of a given point (excluding
the point).
⃗⃗⃗⃗⃗ is the Union of point A and the half line containing all points on the line 𝐴𝐵
⃡⃗⃗⃗⃗ that lie on the
2. The ray 𝐴𝐵
same side of A as point B.
3. Two given points on a line and the portion of the line containing all the points between two points
constitute a line segment. Like ̅̅̅̅
𝐴𝐵 or ̅̅̅̅
𝐵𝐴
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The first letter used to denote a ray names the endpoint; the second letter names some point on the half
line and indicates direction.
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Opposite rays: two rays that define a line with only one point in common are called opposite rays.
Planes
Properties of Planes: a plane can be uniquely determined by four combinations of points and / or lines.
1. Three noncollinear points: Points A, B, and C determine plane 𝛼
2. A line and a point not on that line: ⃡⃗⃗⃗⃗
𝐴𝐵 and C determine plane 𝛼
⃡⃗⃗⃗⃗ and 𝐶𝐷
⃡⃗⃗⃗⃗ determine plane 𝛼
3. Two intersecting lines: 𝐴𝐵
4. Two parallel lines: : ⃡⃗⃗⃗⃗
𝐴𝐵 and ⃡⃗⃗⃗⃗
𝐶𝐷, with ⃡⃗⃗⃗⃗
𝐴𝐵 parallel to ⃡⃗⃗⃗⃗
𝐶𝐷, determine plane 𝛼
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A line and a point not on that line uniquely determine a plane
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Two intersecting lines uniquely determine a plane
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Two parallel lines uniquely determine a plane
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Edge: line separates a plane into 3 parts: the line and two half planes – points on line do not belong to
either half plane; the line is a boundary or edge for both half planes.