Rafael wrote the statements shown in the chart.
Statement 1 Description If a point lies outside a line, then exactly one plane contains both the line and the point.
Statement 2 Description If two points lie in a plane, then the line joining them lies in that plane.
Which option best classifies Rafael’s statements?
Answer
A:Statement 1 and Statement 2 are theorems because they can be proved.
B:Statement 1 and Statement 2 are postulates because they are true facts.
C:Statement 1 is a theorem because it can be proved and Statement 2 is a postulate because it is a true fact.
D:Statement 1 is a postulate because it is a true fact and Statement 2 is a theorem because it can be proved.
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The correct answer is:
C: Statement 1 is a theorem because it can be proved, and Statement 2 is a postulate because it is a true fact.
To classify Rafael's statements, we need to understand the difference between theorems and postulates.
A postulate, also known as an axiom, is a statement that is accepted as true without proof. It serves as the foundation for further mathematical reasoning and can be thought of as a basic or fundamental assumption.
On the other hand, a theorem is a statement that has been proven using logical reasoning and previously established postulates and theorems.
Now let's analyze Rafael's statements:
Statement 1: "If a point lies outside a line, then exactly one plane contains both the line and the point."
This statement describes a relationship between a point, a line, and a plane. It provides a logical connection between these elements and can be proved based on previously accepted postulates and theorems. Therefore, Statement 1 is a theorem.
Statement 2: "If two points lie in a plane, then the line joining them lies in that plane."
This statement establishes a relationship between points and a line in a plane. It is a straightforward statement of fact that holds true without the need for any proof. Therefore, Statement 2 is a postulate.
Based on the analysis above, the correct classification for Rafael's statements is option C: "Statement 1 is a theorem because it can be proved, and Statement 2 is a postulate because it is a true fact."