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Defined Terms and Postulates
April 3, 2008
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Defined terms
• Yesterday, we talked about undefined terms. Today, we will focus on defined terms (which are based on the undefined terms from yesterday).
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More about space and points
• Space is the set of all points (objects in space have length, width, and height).
• Collinear points are points all in one line.• Coplanar points are points all in one plane.
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Intersections
• The intersection of two figures is the set of points that are in both figures.
• You have intersections between lines, between planes, and between a line and a plane.
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Line segments and betweenness
• A line segment is a part of a line. It has a fixed length and is named for its two endpoints. In the figure below segment AB consists of points A and B and all points that are between A and B.
• C is between A and B.
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Rays
• A ray has one end point and extends in the other direction upto infinity. It is represented by naming the end point and any other point on the ray.
• Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line.
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More about segments
• The length a segment is always measured positively, like a distance.
• Congruent segments are segments that have equal lengths.
• The midpoint of a segment is the point that divides the segment into two congruent segments.
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Even more about segments
• A bisector of a segment is a line, segment, ray or plane that intersects the segment at its midpoint.
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About angles
• An angle is the figure formed by two rays with the same endpoint.
• This shared endpoint is called the vertex.• The two rays are called the sides of the angle.• We can name the angle B, angle ABC, angle
CBA, or angle 1.
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Naming angles
• Find several ways to name the two angles.
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Measuring angles
• Angles are measured in degrees. Like distance, degrees are positive.
• In geometry we measure angles in degrees and angles can have degree measures from 0° to 180°.
• Angles are classified according to their measures.– Acute angle: Measure between 0 and 90– Right angle: Measure 90– Obtuse angle: Measure between 90 and 180– Straight angle: Measure 180
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Congruent Angles and Adjacent Angles
• Congruent angles are angles that have equal measures.
• Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.
• In essence it tells us that if the angles "overlap" then we cannot call them adjacent.
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Making conclusions
• Do not assume anything that a diagram does not tell you!
• Notice the marks for congruence in line segments and angles, as well as the mark for a 90 degree angle.
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What is a postulate?
• Postulates or axioms are known as basic assumptions. Assumptions are statements accepted without proof.
• Some will seem very basic.
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Postulate 1 Ruler Postulate
1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1.
2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.
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Postulate 2 Segment Addition Postulate
• If B is between A and C, then AB + BC = AC
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Postulate 3 Protractor Postulate
• On line AB in a given plane , choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with the real numbers from zero to 180 in such a way that:a. OA is paired with 0, and OB with 180.b. If OP is paired with x, and OQ with y, then the measure of POQ = | x - y |.
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Postulate 4 Angle Addition Postulate
• If point B lies in the interior of angle AOC, then the measure of angle AOB + the measure of angle BOC = the measure of angle AOC.
• If angle AOC is a straight angle and B is any point not on AC, then the measure of angle AOB + the measure of angle BOC = 180.
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Postulate 5
• A line contains at least two points; a plane contains at least 3 points not all in one line; space contains at least four points not all in one plane.
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Postulate 6
• Through any two points there is exactly one line.
• True for plane geometry.
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Postulate 7
• Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.
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Postulate 8
• If two points are in a plane, then the line that contains the points is in that plane.
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Postulate 9
• If two planes intersect, then their intersection is a line.