Geometry starts with the basics - plane geometry, which is the study of lines and shapes in two dimensions. From that foundation, geometry constructs increasingly complex models to more accurately portray the real world. Three-dimensional, or solid geometry is almost as simple as plane geometry, with the added dimension of depth.
The building blocks for geometric forms are lines and angles. Understanding the meanings of these terms is an important part of solving problems on the GMAT.
Line: A straight path of points that extends forever in two directions. A line doesn’t have any width or thickness. Arrows are sometimes used to show that the line goes on forever.
Line segment: The set of points on a line between any two points on the line. Basically, it’s just a piece of a line from one point to another that contains those points and all the points between.
Ray: A ray is like half of a line; it starts at an endpoint and extends forever in one direction. You can think of a ray as a ray of light extending from the sun (the endpoint) and shining as far as it can go.
Midpoint: The point halfway (equal distance) between two endpoints on a line segment.
Bisect: To cut something exactly in half, such as when a line, or bisector, cuts another line segment, angle, or polygon into two equal parts.
Intersect: Intersect simply means to cross; that is, when one line or line segment crosses another line or line segment.
Collinear: A set of points that lie on the same line.
Vertical: Lines that run straight up and down.
Horizontal: Lines that run straight across from left to right.
Parallel: Lines that run in the same direction, always remaining the same distance apart. Parallel lines never cross one another.
Perpendicular: When two lines intersect to form a square corner. The intersection of two perpendicular lines forms a right, or 90-degree, angle.
There are some special properties of parallel lines and intersecting lines. The most fundamental aspects of the Geometry emerge from these concepts. Such lines show the relationship between the angles formed.
Parallel lines are the lines in the same plane that, if extended to infinity, would never intersect.
The angles formed by the intersection of a line with two parallel lines have some interesting properties.
When lines intersect, the measures of the angles on opposite sides of the intersection are equal; these angles are called vertical angles. All of the angles labelled x have equal measures, and all of the angles labelled y have equal measures.
The measures of the angles on the same side of a line, represented as x and y, must add up to 180 degrees; these are called supplementary angles.
When parallel lines are cut by a transversal, three important angle relationships are formed.