Question

Determine whether each statement is true or false. If two planes intersect in a line, then the line is in both planes.

Short answer

True, the intersection line belongs to both planes.

Step-by-step solution

Understanding the Problem

To determine if the statement "If two planes intersect in a line, then the line is in both planes" is true or false, we need to analyze what it means for two planes to intersect and what it implies for this line.

Analyzing Plane Intersection

When two planes intersect, they do so along a line. A plane is a flat, two-dimensional surface that extends infinitely in all directions. If they intersect in a line, it means there is a set of points common to both planes.

Defining the Intersection Line

The line formed by the intersection is comprised of points that are a part of both planes. Thus, by definition, any point on this intersection line is present in both planes.

Concluding the Analysis

Since the points on the intersection line are present in both planes, the whole line, being a collection of such points, lies in both planes.

Key concepts

Plane Intersection

When two planes intersect, they share a common line. This might seem a bit abstract at first, but think of two sheets of paper crossing each other. Where they cross is a line running along both sheets. Mathematically, a plane is an infinite two-dimensional surface, and when two such surfaces intersect, they create a line made up of points that exist on both planes.

• Any point on this intersection line belongs to both planes.
• Consider the line like a seam, stitching the two infinite surfaces together.

Therefore, the original statement "If two planes intersect in a line, then the line is in both planes" is true.

Line in Geometry

A line in geometry is a straight one-dimensional figure having no thickness and extending infinitely in both directions. In more understandable terms, think of it as a straight path that never ends. In the context of intersecting planes, a line is very important because it indicates the intersection pathway of the two planes.

• A line has no endpoints and extends forever, regardless of how we depict it on paper.
• It is defined by two distinct points, and these are often called the endpoints when we draw them, but truly they are just indicative.

In the sphere of intersecting planes, this line is literally where the paths of both planes converge, amalgamating their dimensions at one shared line.

Plane Concepts

Understanding a plane requires visualizing something that is perfectly flat and extends infinitely. You might recall the concept of a 'flat Earth' as a humorous analogy, but planes in geometry are quite similar in imagination—except they are truly flat and infinite.

• Planes have no thickness but indefinite width and depth.
• They are often represented by a four-cornered figure, but this is merely a finite representation of an infinite idea.

Functionally, planes serve as a fundamental building block in geometry. Whether they intersect with each other or serve as a canvas for plotting lines and shapes, their infinite nature is central to understanding many multi-dimensional concepts.