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Unit 1 . Foundations of Geometry. Points, Lines, and Planes . Unit 1: Foundations of Geometry. Background.
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Unit 1 Foundations of Geometry
Points, Lines, and Planes Unit 1: Foundations of Geometry
Background • Historically, much of geometry was developed as Euclidean geometry, or non-coordinate geometry. It was named after the Greek mathematician Euclid. Euclid’s most important work was the 13 volumes of The Elements of Geometry. He began his system of geometry with three undefined terms: point, line, and plane. From those terms he defined other geometric vocabulary and postulates. Euclid then proceeded to prove theorems using the definitions and postulates, much as we do today.
Geometric Vocabulary • Undefined Terms: These terms can only be explained using examples and descriptions. These undefined terms can be used to define other geometric terms and properties. (The building blocks of geometry.) • Point • Line • Plane
Point • Description: • Has no actual size, used to represent an abject or location in space. • Naming: • Named by a capital letter. • Symbolic Representation:
Line • Description: • Has no thickness or width, used to represent a continuous set of linear points that extend indefinitely in both directions. • Naming: • Named by a lowercase script letter or by two points on the line. • Symbolic Representation:
Plane • Description: • Has no thickness, width, or depth, used to represent a flat surface that extends indefinitely in all directions. • Naming: • Named by a capital script letter or by three non-collinear points in the plane. • Symbolic Representation:
Defined Terms • All other terms in geometry must be definable and a definition included a category and then a list of critical attributes. • Example: Space - Set of all points, boundless and three-dimensional. • “Set of all points” – is the classification • “Boundless and three dimensional” – are the critical attributes that make this definition different from other definitions
Defined Terms • Space • Set of all points, boundless and three dimensional.
Defined Terms • Collinear • Set of points, that all lie in the same line. Two points are always collinear. Three points must be checked to determine if they are collinear.
Defined Terms • Non-collinear • Set of points, that do not all lie on the same line.
Defined Terms • Coplanar • Set of points, or lines, that lie in the same plane. Three points are always coplanar. Four points must be checked to determine if they are coplanar.
Defined Terms • Non-Coplanar • Set of points, or lines, that do not lie in the same plane.
Defined Terms • Skew Lines • Two non-coplanar lines that do not intersect.
Defined Terms • Parallel Lines • Two coplanar lines that do not intersect (same slope in y = mx +b form).
Defined Terms • Perpendicular Lines • Two coplanar lines that intersect at right angles (opposite reciprocal slopes in y = mx + b form).
Intersections of geometric terms • Two lines intersect at a • point
Intersections of geometric terms • Two planes intersect at a • line
Intersections of geometric terms • A line and a plane intersect at a • point
Points, Lines, and Planes Unit 1: Foundations of Geometry
Distance and Length Unit 1: Foundations of Geometry
Ruler Postulate • Points on a line can be paired with real numbers and the distance between the two points can be found by finding the absolute value of the difference between the numbers. Remember, all distance measures must be
Ruler Postulate • The Ruler Postulate can also be used to find the coordinate of a segment’s endpoint given the other endpoint and the segment’s length.
All About Angles Unit 1: Foundations of Geometry
Angles can be named by… • the vertex point if there are no other angles that could be confused. • three letters with the vertex as the center and the other letters representing points from each side. • a small number if one is given in the angle.
