Math 366 Lecture Notes Section 11.1 Basic Notions (of Geometry)

Transcription

1 Math 366 Lecture Notes Section. Basic Notions (of Geometry) The fundamental building blocks of geometry are points, lines, and planes. These terms are not formally defined, but are described intuitively. (see p. 68). Linear Notions In geometry, a line has no thickness, and it extends forever in two directions. It is determined by two points. Collinear points are points on the same line. (Any two points are collinear but not every three points have to be collinear.) A point B is between points A and C if B A, B C, and B is on the part of the line flanked by A and C. Alternate Definition: Point B is between points A and C if A, B, and C are collinear and the sum of the distances from A to B and B to C equals the distance from A to C. A line segment is a subset of a line that contains two points of the line and all points between those two points. Notation: AB or BA. A ray is a subset of a line that contains the endpoint and all points on the line on one side of the point. Notation: AB Planar Notions A plane has no thickness, and it extends indefinitely in two directions. A plane is determined by three points that are not all on the same line. In other words, given three noncollinear points, a unique plane is determined. Points in the same plane are coplanar. Noncoplanar points cannot be placed in a single plane. Lines in the same plane are coplanar lines. Skew lines are lines that do not intersect, and there is no plane that contains them. Intersecting lines are two coplanar lines with exactly one point in common. Concurrent lines are lines that contain the same point. Two distinct coplanar line m and n that have no points in common are parallel lines. Notation: m // n.

2 How many different lines can be drawn through two points? Can skew lines be parallel? Why or why not? On a globe, a line is a great circle, that is, a circle the same size as the equator. How many different lines can be drawn through two different points on a globe? Four Axioms of Euclid 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. Three Theorems of Euclid A line and a point not on the line determine a plane. Two parallel lines determine a plane. Two intersecting lines determine a plane. Find the number of lines determined by 8 points, no 3 of which are collinear. Polya s Four Step Problem-Solving Process (pp. 4, 6-8). Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Look back. 2

3 Other Planar Notions Two distinct planes either intersect in a line or are parallel. (see p. 685) A line and a plane can be related in one of three ways: If a line and a plane have no point in common, the line is parallel to the plane. If two points of a line are in the plane, then the entire line containing the points is contained in the plane. If a line intersects the plane but is not contained in the plane, it intersects the plane at only one point. If a line is contained in a plane, it separates the plane into two half-planes. The plane is the union of three disjoint sets: the two half-planes, and the line that separates them. Angles When two rays share an endpoint, an angle is formed. The rays of an angle are the sides of the angle, and the common endpoint is the vertex of the angle. Notation: ABC, CBA, B. Adjacent angles share a common vertex and a common side and do not have overlapping interiors. 3

4 Angle Measurement An angle is measured according to the amount of opening between its sides. The degree is commonly used to measure angles. A complete rotation about a point has a measure of 360 degrees, written 360. One degree is of a complete rotation. 360 A degree is subdivided into 60 equal parts, minutes, and each minute is divided into 60 equal parts, seconds. The measuring device pictured is a protractor. How to use protractor: Note: The degree is not the only unit used to measure angles. Radians are sometimes used in calculus and science; its measure is the angle whose arc length is the same as the radius (approximately 57 ). Grads are sometimes used in civil engineering; its measure is of 90, 00 or.9. Convert 8.42 to degrees, minutes, and seconds. Types of Angles (see pp. 689) An acute angle has measure less than 90. A right angle has measure 90. An obtuse angle has measure between 90 and 80. A straight angle has measure 80. Note: In higher mathematics and in scientific applications, it is important to view an angle as being created by a ray rotating about its endpoint. If the ray makes one full rotation, we say that is sweeps an angle of 360. Angles with positive measure are created by a counterclockwise rotation; angles with negative measure by a clockwise rotation. Angles whose measure is greater than 360 are created when the ray makes more than one full rotation. 4

5 Perpendicular Lines When two lines intersect so that the angles formed are right angles, the lines are perpendicular lines. Two intersecting segments and/or rays are perpendicular if they lie on perpendicular lines. Notation: m n, AB AC, etc. A Line Perpendicular to a Plane A line perpendicular to a plane is a line that is perpendicular to every line in the plane through its intersection with the plane. 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? Is it possible for a line intersecting a plane to be perpendicular to two distinct lines in a plane going through its point of intersection with the plane, and yet not be perpendicular to the plane? Can a line be perpendicular to infinitely many lines? Why or why not? List the possible situations for two distinct lines. List the possibilities for two lines in a plane. List the possibilities for two distinct lines in a plane. 5