Question

select all the statements that are postulates.
a) a plane contains at least three noncollinear points.
b) if two points lie in a plane, then the entire line containing those points lies in that plane.
c) if c is the midpoint of \\(\overline{ab}\\), then \\(\overline{ac} \cong \overline{cb}\\).
d) through any two points, there is exactly one line.
e) if two lines intersect, then their intersection is exactly one point.
f) if two angles are vertical angles, then they are congruent.

Explanation:

• Option A: A plane is defined by at least three noncollinear points (postulate), so this is a postulate.
• Option B: If two points lie in a plane, the line through them lies in the plane (postulate), so this is a postulate.
• Option C: If \( C \) is the midpoint of \( \overline{AB} \), then \( \overline{AC} \cong \overline{CB} \) (definition of midpoint, not a postulate).
• Option D: Through any two points, there is exactly one line (postulate), so this is a postulate.
• Option E: If two lines intersect, their intersection is exactly one point (postulate), so this is a postulate.
• Option F: If two angles are vertical angles, they are congruent (theorem, not a postulate).

Answer:

A. A plane contains at least three noncollinear points,
B. If two points lie in a plane, then the entire line containing those points lies in that plane,
D. Through any two points, there is exactly one line,
E. If two lines intersect, then their intersection is exactly one point