Lara wrote the statements shown in the chart.
Statement One: If two lines intersect, then they intersect at exactly one point
Statement Two: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs.
Which option best classifies Lara’s statements?
Answer
A: Statement 1 is a postulate and Statement 2 is a theorem.
B: Statement 1 is a theorem and Statement 2 is a postulate
C: Statement 1 and Statement 2 are theorems because they can be proved.
D: Statement 1 and Statement 2 are postulates because they are true facts.
To determine which option best classifies Lara's statements, we need to understand the definitions of postulates and theorems.
A postulate, also known as an axiom, is a statement that is assumed to be true without proof. It serves as a basic starting point for reasoning and proving other mathematical statements.
A theorem, on the other hand, is a statement that can be proven using logical reasoning and previously established facts, such as other theorems, postulates, or definitions.
Let's analyze Statement One: "If two lines intersect, then they intersect at exactly one point." This statement describes the basic properties of intersecting lines and can be considered a postulate. It is a fundamental assumption that helps build the foundation of Euclidean geometry. Therefore, Statement One is a postulate.
Moving on to Statement Two: "In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length of the legs." This statement is known as the Pythagorean theorem, which is a well-known result in mathematics. The Pythagorean theorem has been proven using deductive reasoning, making it a theorem.
Based on our analysis, the correct option is A: Statement 1 is a postulate, and Statement 2 is a theorem.