Solved on Jan 08, 2024
Determine if two planes always, sometimes, or never intersect in a line. Explain. sometimes, two planes can intersect in a line or single point .
STEP 1
Assumptions
1. We are working within the context of Euclidean geometry.
2. Planes are considered to be infinitely extending flat surfaces.
3. The term "intersect" refers to the set of all points that two figures have in common.
STEP 2
Understand the definition of a plane in Euclidean geometry.
A plane is defined as a flat, two-dimensional surface that extends infinitely in all directions.
STEP 3
Consider the different ways in which two planes can be oriented with respect to each other.
1. Two planes can be parallel, meaning they do not intersect at any point.
2. Two planes can be coincident, meaning they lie on top of each other and thus have all points in common.
3. Two planes can intersect each other.
STEP 4
Understand the definition of intersection in the context of planes.
The intersection of two geometric figures is the set of points that are common to both figures.
STEP 5
Analyze the case where two planes are parallel.
If two planes are parallel, they do not intersect at any point, and therefore they do not intersect in a line.
STEP 6
Analyze the case where two planes are coincident.
If two planes are coincident, they coincide with each other completely, and every point on one plane is also on the other plane. This is not an intersection in a line.
STEP 7
Analyze the case where two planes are not parallel and not coincident.
If two planes are not parallel and not coincident, they will intersect in a line. This line is the set of all points that are common to both planes.
STEP 8
Conclude the possible scenarios for the intersection of two planes.
Based on the analysis, two planes can either not intersect at all (if they are parallel), intersect in a line (if they are not parallel and not coincident), or coincide completely (if they are coincident).
STEP 9
Determine the correct answer to the problem.
The statement that two planes always intersect in a line is incorrect because they may be parallel and not intersect at all.
The statement that two planes never intersect in a line is incorrect because they can intersect in a line if they are not parallel.
The statement that two planes sometimes intersect in a line is correct because it accounts for the possibility that they may be parallel and not intersect, or they may intersect in a line if they are not parallel.
STEP 10
Provide the correct answer to the problem.
The correct answer is "sometimes; Two planes that are not parallel intersect in a line."
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