Basic Geometric Concepts and Definitions
Magnitude
Magnitude: A property of objects, considered individually with respect to other properties it may have. It is required to be measurable, in the sense that it is possible to represent it by numbers.
Extensive vs. Intensive Magnitudes
- Extensive: It is possible to define the sum between different quantities of that magnitude, and that operation verifies certain properties (opposite element, associative, commutative, neutral element) (e.g., length, speed).
- Intensive: The sum operation does not make sense (e.g., color, intensity of an earthquake).
Fundamental Geometric Elements
Plane
Plane: A flat surface that extends without edges (infinite plane).
Line
Line: A long mark of very slight thickness without end in both directions (straight line: with no change of direction).
Point
Point: What cannot be divided into parts.
Semiplane
Semiplane: Each one of the parts into which a plane is divided by a straight line.
Ray
Ray: Each one of the points into which a line is divided by a point. This point is called the origin.
Segment
Segment: A part of a line between two points. These points are the extremes of the segment.
Types of Segments
- Successive Segments: Two or more segments verifying that the origin of one segment is the end point of another.
- Consecutive Segments: Successive segments, all of them on the same straight line.
Angles
Angles: Two rays with the same origin that separate the plane into two infinite regions.
Types of Angles
- Concave: The region that contains the prolongation of the sides of the angle.
- Convex: The region that does not contain the prolongation of the sides of the angle.
Angles Depending on Their Location
- Adjacent Angles: Angles sharing the same vertex and a common side, verifying that the non-common sides are in different semiplanes with respect to the common side.
- Adjacent Angles on a Straight Line: Adjacent angles verifying that the non-common sides are in the same straight line; the non-common sides are in different semiplanes.
- Opposite Angles: Angles sharing the same vertex, where the sides of one angle are the prolongation of the sides of the other angle.
Angles Depending on Their Opening
- Straight Angle: Angle having both sides in the same straight line. One side is the prolongation of the other side.
- Right Angle: Each one of the two equal adjacent angles on a straight line.
- Acute Angle: An angle with the openness of its sides smaller than in the right angle.
- Obtuse Angle: Angle with the openness of its sides smaller than in the straight line and bigger than in the right angle.
- Null Angle: Angle with coincidental sides that is convex (0º).
- Total Angle: Angle with coincidental sides that is concave (360º).
Complementary and Supplementary Angles
- Complementary Angles: Two angles whose addition is 90º.
- Supplementary Angles: Two angles whose addition is 180º. All adjacent angles on a straight line are supplementary, but not vice versa because they don’t need to share the vertex.
Parallel and Perpendicular Lines
Parallel Lines
Parallel: Two different lines are parallel if, being in the same plane, they do not share any point.
Perpendicular Lines
Perpendicular: Two straight lines are perpendicular if they intersect, creating four right angles.
Bisectors
Bisector of an Angle
Bisector of an Angle: A ray with its origin in the vertex that divides the angle into two equal angles. All the points in the bisector are at the same distance from both sides of the angle.
Perpendicular Bisector
Perpendicular Bisector: A straight line that divides a segment into two equal parts and is perpendicular to the segment. All the points in the perpendicular bisector are at the same distance from both extremes of the segment.
Angles Formed by Parallel Lines and a Transversal
- Interior Angles: Angles between the parallel lines.
- Exterior Angles: Angles that are not interior.
- Corresponding Angles: Angles that are in the same position on each line (measure the same).
- Alternate Interior Angles: Angles on the opposite sides of the transversal line and on the interior of the parallel lines, in different vertices (measure the same).
- Alternate Exterior Angles: Angles on the opposite sides of the transversal and on the exterior of the parallel lines, in different vertices (measure the same).
Polygonal Path
Polygonal Path: A set of successive segments. It can be:
- Opened: Polygonal path with free extremes.
- Closed: Polygonal path with no free extremes.