To determine which of the given statements are theorems, let's analyze each one:
Statement #1: All triangles have three sides and three angles.
This statement defines the fundamental characteristics of a triangle. By definition, a triangle is a polygon with three edges and three vertices. Thus, this statement is a definition, not a theorem.
Statement #2: The interior angles of any triangle sum to 180 degrees.
This statement asserts a well-known geometric property of triangles. It is derived from Euclidean geometry and can be proven using various methods, such as by considering the parallel postulate or by dividing the triangle into two right triangles. Since it is a proven statement within geometry, it is classified as a theorem.
Statement #3: [tex]\(a + b = b + a\)[/tex].
This statement refers to the Commutative Property of Addition. It is a fundamental property of arithmetic that can be proven within the axioms of number theory. Therefore, while it is a fundamental property, it is generally regarded as an axiom or property, not a theorem.
Based on this analysis:
- Statement #1 is not a theorem; it is a definition.
- Statement #2 is a theorem; it states a geometric property that has been proven.
- Statement #3 is not a theorem; it is a fundamental property of arithmetic.
In summary, the only statement that is a theorem is Statement #2. The results are:
- Statement #1: Not a theorem.
- Statement #2: Is a theorem.
- Statement #3: Not a theorem.
