Geometric Structure Basic Information

IntroductionInstructionExamplesPractice Geometry is a way of thinking about and seeing the world. Geometry is evident in nature, art and culture. What geometric objects do you see in this picture?

IntroductionInstructionExamplesPractice Geometry is both ancient and modern. Geometry originated as a systematic study in the works of Euclid, through its synthesis with the work of Rene Descartes, to its present connections with computer and calculator technology. What geometric objects do you see in this picture?

IntroductionInstructionExamplesPractice The basic terms and postulates of geometry will be introduced as well as the tools needed to explore geometry. What geometric objects do you see in this picture?

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IntroductionInstructionExamplesPractice points linesplanes Three building blocks of geometry are points, lines and planes. They are considered building blocks because they are basic and not defined in terms of other figures. Even though the terms cannot be defined, they may be described and depicted with a model. This is page 1 of 22 Page list LastNext Many-Sided World of Geometry, The, Program 1: Geometry Basics. Standard Deviants (2002). Retrieved July 12, 2006, from unitedstreaming:

IntroductionInstructionExamplesPractice pointA point is the most basic building block of geometry. A point has no size. A point indicates location. A dot represents or models a point. A point is named with a capital letter. This is page 2 of 22 Page list LastNext

IntroductionInstructionExamplesPractice lineA line is a straight, continuous arrangement of infinitely many points. A line has infinite length but no thickness. A line extends forever in two directions. A line may be named with two identified points on the line. A line symbol (double-headed arrows) is placed over the letters naming the points on the line. Alternatively, line may be named with a single, lower case letter. This is page 3 of 22 Page list LastNext

IntroductionInstructionExamplesPractice planeA plane has length and width but no thickness. A plane is like a flat surface the extends infinitely along its length and width. Now, click the Sketchpad logo to demonstrate this description. A plane is represented line extends with four-sided figure, like a tilted piece of paper, drawn in perspective. A plane is named with a script capital letter, Q. This is page 4 of 22 Page list LastNext

IntroductionInstructionExamplesPractice In the figure at the right, the flat surface represents a portion of plane Q. planeA plane may also be named using three points that lie in the plane, such as G, F and E; please note that the three points are not on the same line. This is page 5 of 22 Page list LastNext

IntroductionInstructionExamplesPractice Collinear pointsCollinear points are points that lie on the same line. In the figure at the right, A, B and C are collinear. A, B and D are noncollinear (not collinear). Any two points are collinearAny two points are collinear. This is page 6 of 22 Page list LastNext

IntroductionInstructionExamplesPractice Coplanar pointsCoplanar points are points that lie in the same plane. In the figure at the right, E, F, G, and H are coplanar. E, F, G, and J are noncoplanar (not coplanar). Any three points are coplanar.Any three points are coplanar. This is page 7 of 22 Page list LastNext

IntroductionInstructionExamplesPractice With the foundational terms (point, line and plane) described, other geometric figures may be defined. This is page 8 of 22 Page list LastNext “Let no one ignorant of geometry enter my door.” - Plato

IntroductionInstructionExamplesPractice segmentA segment is a part of a line that begins at one point and ends at another. endpointsThe points are called the endpoints of the segment. A segment is named by its endpoints. A bar (no arrows) is drawn over the two capitalized letters. This is page 9 of 22 Page list LastNext

IntroductionInstructionExamplesPractice ray is aA ray is a part of a line that starts at a point and extends infinitely in one direction. endpointThe point is called the endpoint of the ray. A ray is named its endpoint first. A single arrow (toward the infinite direction) is drawn over the two capitalized letters. This is page 10 of 22 Page list LastNext

IntroductionInstructionExamplesPractice angleAn angle is a figure formed by two rays with a common endpoint. vertex of the angleThe common endpoint is the vertex of the angle. sides of the angleThe rays are the sides of the angle. Angles are formed when lines, rays, or line segments intersect. This is page 11 of 22 Page list LastNext

IntroductionInstructionExamplesPractice An angle divides the plane into two regions Interior Exterior If two points, one from each side of the angle, are connected with a segment, the segment passes through the interior of the angle. This is page 12 of 22 Page list LastNext

IntroductionInstructionExamplesPractice An angle is named using three points. The vertex must be the middle point of the name. Write SRT or TRS. Say “angle S R T” or “angle T R S.” If there is no possibility of confusion, the angle may be named S or 1. This is page 13 of 22 Page list LastNext

IntroductionInstructionExamplesPractice intersectWhen geometric figures have one or more points in common, they are said to intersect. The set of points that they have in common is called their intersection. This is page 14 of 22 Page list LastNext

IntroductionInstructionExamplesPractice axiomatic systemAn axiomatic system is a set statements. axiomspostulatesThe statements are axioms or postulates; these statements are accepted without proof. theoremsThe other statements in an axiomatic system include theorems; theorems are truths that can be derived from postulates. This is page 15 of 22 Page list LastNext

IntroductionInstructionExamplesPractice Mathematicians accepts undefined terms and definitions so that a consistent system may be built. Like the pieces of a model that rest on others, the theorems of an axiomatic system rest on postulates and other theorems. The system must be arranged logically. This is page 16 of 22 Page list LastNext

IntroductionInstructionExamplesPractice As with all axiomatic systems, geometry is connected with logic. proofThis logic is typically expressed with convincing argument or proof. Proof may be used to demonstrate that things are not true. This is page 17 of 22 Page list LastNext

IntroductionInstructionExamplesPractice Examine the geometric model at the right. Specifically, identify the places where lines intersect each other. Complete the postulate: The intersection of two lines is a ___________. This is page 18 of 22 Page list LastNext point

IntroductionInstructionExamplesPractice Consider the model. Specifically, identify the places in the diagram where planes intersect each other. Complete the postulate: The intersection of two planes is a _______________. This is page 19 of 22 Page list LastNext line

IntroductionInstructionExamplesPractice Consider the model. Look at points A and E. How many lines pass through these two points? Complete the postulate: Through any two points there is exactly one ______________. This is page 20 of 22 Page list LastNext line

IntroductionInstructionExamplesPractice Consider the model. Look at points A, E and H. How many planes pass through these three noncollinear points? Complete the postulate: Through any three noncollinear points there is exactly one _______________. This is page 21 of 22 Page list LastNext plane

IntroductionInstructionExamplesPractice Pick any plane in the illustration; pick two points that are in the plane. Name the line that passes Is the line in the plane you selected? Complete the postulate: If two points are in a plane, then the line containing them ________________. This is page 22 of 22 Page list LastNext is in the plane

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IntroductionInstructionExamplesPractice Example 1 Example 2 Examples Can you answer questions 1-8 and 12-14?

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IntroductionInstructionExamplesPractice Holt On-line & Homework help Lesson 1.1 What geometric objects do you see in this picture?

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Example 1 Back to main example page Name the intersection of plane ABDC and plane YZDB. How many lines in the figure contain point Z? How many planes in the figure contain line BY? True or false: Two planes intersect in exactly one point. line BD Two – line DZ and YZ plane BAXY and plane YZDB False - line 2

Example 2 truefalse Classify each statement as true or false. Two lines intersect in a plane. Two lines are contained in exactly one plane. Any three points are contained in exactly one line. Back to main example page False - point False – could be skew lines False – only collinear points

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