Plane Geometry: Points, Lines, Angles, and Shapes

By Mark Zegarelli

Plane geometry is the study of figures on a two-dimensional surface — that is, on a plane. You can think of the plane as a piece of paper with no thickness at all. Technically, a plane doesn’t end at the edge of the paper — it continues forever.

Making some points

A point is a location on a plane. It has no size or shape. Although in reality a point is too small to be seen, you can represent it visually in a drawing by using a dot. When two lines intersect, they share a single point. Additionally, each corner of a polygon is a point.

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Knowing your lines

A line — also called a straight line — is pretty much what it sounds like; it marks the shortest distance between two points, but it extends infinitely in both directions. It has length but no width, making it a one-dimensional (1-D) figure.

Given any two points, you can draw exactly one line that passes through both of them. In other words, two points determine a line.

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When two lines intersect, they share a single point. When two lines don’t intersect, they are parallel, which means that they remain the same distance from each other everywhere. A good visual aid for parallel lines is a set of railroad tracks. In geometry, you draw a line with arrows at both ends. Arrows on either end of a line mean that the line goes on forever.

A line segment is a piece of a line that has endpoints.

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A ray is a piece of a line that starts at a point and extends infinitely in one direction, kind of like a laser. It has one endpoint and one arrow.

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Figuring the angles

An angle is formed when two rays extend from the same point.

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Angles are typically used in carpentry to measure the corners of objects. They’re also used in navigation to indicate a sudden change in direction. For example, when you’re driving, it’s common to distinguish when the angle of a turn is “sharp” or “not so sharp.”

The sharpness of an angle is usually measured in degrees. The most common angle is the right angle — the angle at the corner of a square — which is a 90-degree angle:

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Angles that have fewer than 90 degrees — that is, angles that are sharper than a right angle — are called acute angles, like this one:

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Angles that measure greater than 90 degrees — that is, angles that aren’t as sharp as a right angle — are called obtuse angles, as seen here:

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When an angle is exactly 180 degrees, it forms a straight line and is called a straight angle.

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Shaping things up

A shape is any closed geometrical figure that has an inside and an outside. Circles, squares, triangles, and larger polygons are all examples of shapes.

Much of plane geometry focuses on different types of shapes.