Basic Geometrical Ideas - Angles, Vertices, and Sides

Point and Line Segment: A point is a tiny dot that marks a specific location in space and has no dimensions. A line segment is the shortest path between two points, representing a straight line with fixed endpoints. Visually, a line segment $\overline{AB}$ looks like a straight bar connecting dots $A$ and $B$.

Ray: A ray is a part of a line that starts at a fixed point (called the initial point) and goes on forever in one direction. Visually, it is represented as a line with a dot at one end and an arrow at the other, such as $\overrightarrow{OA}$ where $O$ is the starting point.

The Concept of an Angle: An angle is formed when two rays originate from a common starting point. This common point is called the vertex. Imagine the two hands of a clock joined at the center; the space or 'opening' between the hands represents the angle.

Vertex and Arms: In an angle, the shared starting point of the two rays is the Vertex. The two rays themselves are referred to as the Arms or Sides of the angle. For an angle formed by rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$, $B$ is the vertex and $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are the arms.

Naming an Angle: To name an angle, we typically use three capital letters where the middle letter is always the vertex. If the vertex is $O$ and points $X$ and $Y$ are on the arms, the angle is named $\angle XOY$ or $\angle YOX$.

Interior and Exterior of an Angle: An angle divides a plane into two regions. The Interior is the region 'inside' the arms of the angle (like the space inside a slice of cake). The Exterior is the entire region 'outside' the arms of the angle.

Vertices and Sides in Polygons: In a closed flat shape made of line segments (a polygon), each line segment is called a Side. The point where any two sides meet is called a Vertex. For instance, a triangle has $3$ sides and $3$ vertices, appearing as three straight edges meeting at three corners.