which statement is not one of the axioms of euclidean geometry?
a. given any two distinct points, there is exactly one line that contains them.
b. every plane contains at least three points that do not lie on the same line.
c. if two points lie in a plane, the line containing these points also lies in the plane.
d. if two planes intersect, their intersection is a point.

Question

which statement is not one of the axioms of euclidean geometry?
a. given any two distinct points, there is exactly one line that contains them.
 b. every plane contains at least three points that do not lie on the same line.
 c. if two points lie in a plane, the line containing these points also lies in the plane.
 d. if two planes intersect, their intersection is a point.

Accepted Answer

# Brief Explanations:
To determine which statement is not an axiom of Euclidean geometry, we analyze each option:
- **Option A**: This is the "two - point postulate", a fundamental axiom of Euclidean geometry. It states that given any two distinct points, there is exactly one line that contains them.
- **Option B**: In Euclidean geometry, a plane is defined such that if there are three non - collinear points (points not on the same line), they determine a unique plane. But the statement "Every plane contains at least three points that do not lie on the same line" is not an axiom. The axioms of Euclidean geometry related to planes focus on the relationship between points, lines, and planes in terms of containment (e.g., if two points lie in a plane, the line containing them lies in the plane) rather than mandating a minimum number of non - collinear points in a plane as an axiom.
- **Option C**: This is the "plane - line postulate", an axiom. It says that if two points lie in a plane, the line containing these points also lies in the plane.
- **Option D**: This is the "plane - intersection postulate", an axiom. It states that if two planes intersect, their intersection is a line (not a point, so this statement is incorrect in the sense that it's misstating the axiom? Wait, no, re - reading: Wait, the option D says "If two planes intersect, their intersection is a point." But the correct axiom is that the intersection of two planes is a line. Wait, but the question is which is NOT an axiom. Wait, maybe I misread option B. Wait, let's re - check:

Wait, the standard Euclidean geometry axioms (postulates) are:

1. A straight line segment can be drawn joining any two points. (Related to option A: Given two distinct points, there is exactly one line containing them)
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (Parallel postulate)

And the postulates about planes:

- A plane contains at least three non - collinear points (points not on the same line). Wait, option B says "Every plane contains at least three points that do not lie on the same line." That is actually a postulate (axiom) of Euclidean geometry for planes. Wait, then what about option D: "If two planes intersect, their intersection is a point." The correct postulate is that the intersection of two planes is a line. So option D is a false statement and not an axiom. Wait, but maybe I made a mistake. Wait, let's check the options again:

Option A: "Given any two distinct points, there is exactly one line that contains them." – Axiom (two - point postulate)

Option B: "Every plane contains at least three points that do not lie on the same line." – Axiom (plane postulate: a plane is determined by three non - collinear points, so every plane must have at least three non - collinear points)

Option C: "If two points lie in a plane, the line containing these points also lies in the plane." – Axiom (plane - line postulate)

Option D: "If two planes intersect, their intersection is a point." – The correct postulate is that the intersection of two planes is a line. So this statement is not an axiom (it's actually false, and the correct axiom is that the intersection is a line). Wait, but the question is which is NOT an axiom. So the answer should be D? Wait, no, maybe I misread option B. Wait, maybe the original question has a typo, but according to the options, let's re - evaluate:

Wait, the key is to recall the Euclidean geometry axioms (postulates) for planes and lines:

- Line - point postulate: Through any two points, there is exactly one line (Option A is this postulate, so it's an axiom)
- Plane - line postulate: If two points lie in a plane, the line containing them lies in the plane (Option C is this, so it's an axiom)
- Plane - intersection postulate: If two planes intersect, their intersection is a line (Option D says it's a point, so this is not an axiom)
- Plane - point postulate: A plane contains at least three non - collinear points (Option B is this, so it's an axiom)

So the statement that is NOT an axiom of Euclidean geometry is D. Wait, but the option D says "If two planes intersect, their intersection is a point." which is incorrect (the correct is a line), so it's not an axiom.

# Answer:
D. If two planes intersect, their intersection is a point.

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