The correct answer is (a):

"The line AB is perpendicular to the line AC if and only if there exists a point C' with $C * A * C'$, $AC \cong AC'$, and $BC \cong BC'$."

Explanation:

• Perpendicularity in geometry typically means that two lines meet at a right angle.
• This definition can be framed in terms of congruence and betweenness.
• The condition $C * A * C'$ expresses betweenness, meaning that point $A$ lies between points $C$ and $C'$.
• The conditions $AC \cong AC'$ and $BC \cong BC'$ express congruence, meaning that the segments $AC$ and $AC'$ are of equal length, and similarly for $BC$ and $BC'$.
• These conditions effectively describe the idea that point $B$ lies at the same distance from both points $C$ and $C'$, thus forming a 90-degree angle between line $AB$ and line $AC$, which is the geometric definition of perpendicularity.

The other options are incorrect because:

• (b) and (c) place too strict conditions by saying "for all $C'$" instead of specifying the existence of one such $C'$, which complicates the definition unnecessarily.
• (d) is incorrect because the notion of perpendicularity can be defined in terms of congruence and betweenness, as shown in option (a).

Follow-up questions:

1. What does betweenness mean in the context of geometric points?
2. How is congruence defined for line segments in geometry?
3. Why are the conditions $AC \cong AC'$ and $BC \cong BC'$ important for perpendicularity?
4. Can perpendicularity always be expressed in terms of congruence and betweenness in non-Euclidean geometry?
5. What is the relationship between perpendicularity and the Pythagorean theorem?

Tip:

When proving geometric properties, consider using basic definitions like congruence and betweenness, as they often provide a clear path to the solution.