Certainly! Let's analyze each statement to determine which one is a theorem.
Statement #1: All triangles have three sides and three angles.
This statement describes the fundamental definition of a triangle. By definition, a triangle is a polygon with three sides and three angles. So, this statement is a definition, not a theorem.
Statement #2: The interior angles of any triangle sum to 180 degrees.
This statement tells us about a property of triangles in Euclidean geometry. It states that no matter what type of triangle you have (scalene, isosceles, or equilateral), the sum of the interior angles will always be 180 degrees. This is a well-known property that has been proven and is thus a theorem in Euclidean geometry.
Statement #3: [tex]\( a + b = b + a \)[/tex]
This statement is an example of the commutative property of addition, which states that the order in which two numbers are added does not affect the sum. This property is considered an axiom or basic truth in algebra. Axioms are accepted without proof and serve as a starting point for further reasoning and arguments, so this statement is an axiom, not a theorem.
To summarize:
- Statement #1 is a definition.
- Statement #2 is a theorem.
- Statement #3 is an axiom (the commutative property of addition).
Thus, the statement that is a theorem is Statement #2: The interior angles of any triangle sum to 180 degrees.
