Basic geometric terms and constructions Knowledge-Based Study

Study notes

1. Geometry is built upon fundamental, undefined terms that provide the framework for defining shapes, angles, and spatial relationships. The three primary building blocks are points, lines, and planes.

   1. Point

   A point represents an exact location in space. It has no length, width, or depth (zero dimensions). A point is typically represented by a small dot and named using a single capital letter, such as point A or point P.

   2. Line

   A line is a straight path of points that extends infinitely in two opposite directions. It has length but no width or thickness (one dimension). Between any two distinct points, there is exactly one unique line. A line passing through points A and B can be written as line AB or denoted symbolically as $\overleftrightarrow{AB}$.

   3. Line Segment

   A line segment is a measurable portion of a line bounded by two distinct endpoints. It contains those two endpoints and all points on the line between them. A segment with endpoints A and B is written as segment AB or $\overline{AB}$. The length or distance between points A and B is written simply as AB.

   4. Ray

   A ray is a part of a line that begins at a specific initial point (called the endpoint) and extends indefinitely in one direction. When naming a ray, the endpoint must always be stated first. For example, ray AB, denoted as $\overrightarrow{AB}$, starts at endpoint A and extends indefinitely through point B.

   When two or more lines lie in the same flat surface (coplanar), they can interact in specific geometric ways:

   • Intersecting Lines: Two distinct coplanar lines that cross each other at exactly one common point.
   • Parallel Lines: Lines in the same plane that never intersect, no matter how far they are extended. They maintain a constant perpendicular distance from each other. If line m is parallel to line n, we write m || n.
   • Perpendicular Lines: Two intersecting lines that form four right angles (90 degrees each). If line p is perpendicular to line q, we denote this relationship as p $\perp$ q.

   Angles and Collinear Points

   Points that lie along the exact same straight line are called collinear points. An angle is formed by two rays that share a common endpoint called the vertex. Angles are measured in degrees and categorized as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees and less than 180 degrees), or straight (exactly 180 degrees).

   A geometric construction is a method of drawing accurate geometric figures using only two classical, idealized tools:

   • Straightedge: An unmarked ruler used solely to draw straight lines, segments, or rays through given points.
   • Compass: A device used to draw circles or circular arcs and to transfer exact distances without numerical measurement.

   Core Constructions

   1. Copying a Line Segment: Given segment AB, choose an arbitrary starting point C on a working line. Open the compass to match length AB, place the compass needle at point C, and strike an arc intersecting the line at point D. Segment CD is congruent to segment AB.
   2. Constructing a Perpendicular Bisector: To bisect segment AB and form a line perpendicular to it, set the compass width to more than half the length of AB. Draw two intersecting arcs centered at point A (one above and one below the segment), and repeat the same radius setting from point B. Using a straightedge, connect the two arc intersection points. This line divides segment AB into two equal halves at a 90-degree angle.
   3. Bisecting an Angle: Place the compass point at the vertex of the angle and draw an arc that intersects both sides of the angle. From these two intersection points, draw two equal-radius arcs that intersect inside the interior of the angle. Drawing a ray from the vertex through this intersection creates two congruent adjacent angles.