The image shows a multiple-choice question, where each part asks for the classification of different mathematical statements. Here's a breakdown of how the statements might be classified:

(a) Kaitlin wrote a proof about the midpoint of a segment:

• Statement: "The midpoint of a segment is the point that divides the segment into two congruent segments."
• Answer: This is best described as a definition, since it formally describes what a midpoint is.

(b) Yolanda is guessing a statement about rational numbers:

• Statement: Yolanda is guessing a certain statement about rational numbers is true, but there is no proof yet.
• Answer: This is best described as a conjecture, since it is a statement believed to be true but not yet proven.

(c) John read a proof about planes:

• Statement: "Plane" is best described as a(n)...
• Answer: A definition, because "plane" is a fundamental term in geometry and its properties are described in formal definitions.

(d) Juan guessed a statement about prime numbers and found proof:

• Statement: Juan guessed a certain statement about prime numbers is true, and then he found a proof that it is true.
• Answer: This is best described as a theorem, since a conjecture becomes a theorem once it is proven to be true.

(e) Elsa wrote a proof about congruent segments:

• Statement: "Congruent segments are segments that have the same length."
• Answer: This is best described as a definition, as it provides the formal meaning of congruent segments.

Do you want further clarification on any of these classifications, or more details on the reasoning behind them?

Related Questions:

1. What is the difference between a conjecture and a theorem?
2. Why is the concept of a "plane" considered a definition in geometry?
3. Can you give more examples of conjectures that became theorems after proof?
4. What other terms in geometry require formal definitions like "midpoint" or "plane"?
5. How is proving a conjecture different from providing a definition?

Tip:

Understanding the difference between terms like "definition," "theorem," and "conjecture" is key to mastering proof-based mathematics. Always clarify these terms when studying geometry or other formal branches of mathematics.