Which of the following is true:

 

The notion of perpendicularity can be defined in terms of the notions of congruence and betweenness by

(a) the line AB is perpendicular to the line AC if and only if there exists a point C' with C*A*C', AC≡AC',  and BC≡BC';

(b) the line AB is perpendicular to the line AC if and only if for all C' with  C*A*C', we have BC≡BC';

(c)  the line AB is perpendicular to the line AC if and only if for all C' with   AC≡AC', we have BC≡BC';

(d) the notion of perpendicularity cannot be defined in terms of congruence and betweenness.

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.

Which of the following is true: The notion of perpendicularity can be defined in terms of the notions of congruence and betweenness by (a) the line AB is perpendicular to the line AC if and only if there exists a point C' with C*A*C', AC≡AC', and BC≡BC'; (b) the line AB is perpendicular to the line AC if and only if for all C' with C*A*C', we have BC≡BC'; (c) the line AB is perpendicular to the line AC if and only if for all C' with AC≡AC', we have BC≡BC'; (d) the notion of perpendicularity cannot be defined in terms of congruence and betweenness.

Explore how the concept of perpendicularity can be defined using congruence and betweenness. This problem examines geometric relationships, including key concepts such as the Perpendicularity Theorem, the betweenness of points, and the congruence of line segments. Suitable for students in Grades 10-12.

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