Question

question 2 of 10

which of the following are among the five basic postulates of euclidean geometry?

check all that apply.

a. a circle can be drawn with any center and radius.
b. through a point not on a given line, there is exactly one line parallel to the given line.
c. all right angles are equal to one another.
d. all equilateral triangles are also equiangular.

Explanation:

Identify Euclid's five basic postulates

Euclid's five postulates of classical geometry are:

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 the parallel postulate).

Evaluate the given options

• Option A: "A circle can be drawn with any center and radius." This is a modern phrasing of Postulate 3.
• Option B: "Through a 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 the fifth postulate but is not one of Euclid's original five postulates.
• Option C: "All right angles are equal to one another." This is exactly Postulate 4.
• Option D: "All equilateral triangles are also equiangular." This is a provable theorem, not a basic postulate.

Select the correct statements

The statements that are among Euclid's original five basic postulates are A and C.

Answer:

• A. A circle can be drawn with any center and radius. (Correct answer)
• B. Through a point not on a given line, there is exactly one line parallel to the given line.
• C. All right angles are equal to one another. (Correct answer)
• D. All equilateral triangles are also equiangular.