Intro to Geometry: 동영상 및 연습문제

The basic building blocks of geometry include points, lines, rays, line segments, angles, and planes. A point represents an exact location in space and has no length or width. It is labeled with a capital letter, such as point A. A line extends infinitely in both directions, has no endpoints, and no width. It is named by two points on the line, for example, $\overleftrightarrow{\mathrm{AB}}$. A ray starts at one endpoint and extends infinitely in one direction, named with the endpoint first, like $\overrightarrow{\mathrm{AB}}$. A line segment has two endpoints and finite length, denoted as $\overline{\mathrm{AB}}$. An angle is formed by two rays sharing a common endpoint called the vertex, and is named with three points, the vertex in the middle, such as $\angle ABC$. Finally, a plane extends infinitely in two dimensions, having infinite length and width, and is usually labeled with a capital letter like Plane P.

In geometry, naming lines, rays, and line segments follows specific conventions. A line is named using any two points on it, with a double arrow above the letters to indicate it extends infinitely in both directions, for example, $\overleftrightarrow{\mathrm{AB}}$ or $\overleftrightarrow{\mathrm{BA}}$. The order of points does not matter for lines. A ray has one endpoint and extends infinitely in one direction. When naming a ray, the endpoint must be listed first, followed by another point on the ray, with a single arrow above pointing away from the endpoint, such as $\overrightarrow{\mathrm{AB}}$. You cannot reverse the order because the endpoint must come first. A line segment has two endpoints and finite length. It is named by its endpoints with a line segment symbol above, for example, $\overline{\mathrm{AB}}$ or $\overline{\mathrm{BA}}$. The order of points does not affect the name of the segment.

The key differences between a line, ray, and line segment lie in their endpoints and length. A line has no endpoints and extends infinitely in both directions, so its length is infinite. It is represented with arrowheads on both ends. A ray has exactly one endpoint and extends infinitely in one direction, giving it infinite length as well, but it starts at a specific point. A line segment has two endpoints and a finite length, meaning it starts and ends at specific points and does not extend beyond them. Unlike lines and rays, line segments can be measured because their length is a fixed number. All three have no width, only length.

An angle is formed when two rays share a common endpoint called the vertex. The rays are the sides of the angle. To name an angle, you use three points: one point on the first ray, the vertex point in the middle, and one point on the second ray. For example, if the rays are $\overrightarrow{\mathrm{BA}}$ and $\overrightarrow{\mathrm{BC}}$ sharing vertex B, the angle is named $\angle ABC$ or $\angle CBA$. The vertex must always be the middle letter. Sometimes, if the context is clear, the angle can be named simply by its vertex, such as $\angle B$. The size of the angle is determined by the amount of rotation from one ray to the other.

A plane in geometry is a flat, two-dimensional surface that extends infinitely in all directions along its length and width. Unlike lines, which extend infinitely in only one dimension (length), and rays, which extend infinitely in one direction from a point, a plane has infinite length and infinite width. You can think of a plane like a giant whiteboard that stretches forever vertically and horizontally. Planes have no thickness and are usually named with a single capital letter, such as Plane P. They serve as the backdrop for many geometric figures and are fundamental in understanding spatial relationships.