Question

question 11 of 25

which of the following is not a postulate of euclidean geometry?

a. all equilateral triangles have interior angles equal to 60 degrees.
b. a circle can be drawn with any center and radius.
c. through a given point not on a given line, there is exactly one line parallel to the given line.
d. a straight line segment can be drawn between any two points.

Explanation:

Analyze the definition of Euclidean postulates

Euclid's five postulates are the foundational assumptions of Euclidean geometry:

1. A straight line segment can be drawn joining any two points.
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 equal to one another.
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 inevitable must meet each other on that side if extended far enough (equivalent to Playfair's axiom: through a given point not on a given line, there is exactly one line parallel to the given line).

Evaluate the given options

• A: "All equilateral triangles have interior angles equal to 60 degrees." This is a theorem derived from the postulates, not a postulate itself.
• B: "A circle can be drawn with any center and radius." This corresponds directly to Postulate 3.
• C: "Through a given point not on a given line, there is exactly one line parallel to the given line." This is Playfair's Axiom, which is logically equivalent to Euclid's fifth postulate (the Parallel Postulate).
• D: "A straight line segment can be drawn between any two points." This corresponds directly to Postulate 1.

Identify the non-postulate

Option A is a geometric fact (theorem) that must be proven using the postulates, making it the correct choice for what is not a postulate.

Answer:

• (A) All equilateral triangles have interior angles equal to 60 degrees. (Correct answer)
• (B) A circle can be drawn with any center and radius.
• (C) Through a given point not on a given line, there is exactly one line parallel to the given line.
• (D) A straight line segment can be drawn between any two points.