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.
its A
The correct answer is A: Statement 1 is a postulate and Statement 2 is a theorem.
Explanation:
A postulate, also known as an axiom, is a statement that is accepted without proof. It is considered to be true by definition. In this case, Statement 1 is a postulate because it describes a basic property of lines intersecting at a single point.
A theorem, on the other hand, is a statement that can be proven using logical reasoning and previously established truths. Statement 2 is a theorem because it can be proven using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
To determine how to classify Lara's statements, we need to understand the definitions of a postulate and a theorem.
A postulate, also known as an axiom, is a statement that is accepted without proof. It is considered a basic assumption or a self-evident truth.
A theorem, on the other hand, is a statement that can be proven using logical reasoning and previously accepted statements (postulates, theorems, etc.).
Let's evaluate each statement:
Statement One: "If two lines intersect, then they intersect at exactly one point." This statement is a fundamental geometric idea that is accepted without proof. It is not something that can be proven through logical reasoning. Therefore, Statement One qualifies as a postulate.
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 a well-known mathematical relationship in geometry, named the Pythagorean theorem. The Pythagorean theorem can be proven using various methods, such as algebraic manipulation or geometric constructions. Therefore, Statement Two qualifies as a theorem.
Based on this analysis, the correct option is:
A: Statement 1 is a postulate, and Statement 2 is a theorem.