Final answer:
Sets of side lengths for each type of triangle must adhere to their respective definitions and the Triangle Inequality Theorem. Equilateral triangles have three equal sides, isosceles triangles two equal sides, and scalene triangles have all sides of different lengths. A triangle with at least two angles of the same measure could be an equilateral or isosceles triangle.
Step-by-step explanation:
The question is asking to identify the sets of side lengths that can construct various types of triangles. Understanding the definition and properties of these triangles is essential for determining the correct sets of side lengths:
- Equilateral triangle: All three sides are of equal length. For example, side lengths of 5 cm, 5 cm, and 5 cm.
- Isosceles triangle: Two sides are of equal length and the third side is of a different length. An example could be side lengths of 4 cm, 4 cm, and 2 cm.
- Scalene triangle: All three sides have different lengths. An example set of side lengths could be 3 cm, 4 cm, and 5 cm.
- Triangle with at least two angles of the same measure: This could be either an equilateral triangle (all angles are the same since all sides are the same), or an isosceles triangle (which has at least two angles of the same measure due to two sides being equal).
To ensure that a set of side lengths can form a triangle, the Triangle Inequality Theorem must be satisfied; the sum of the lengths of any two sides must be greater than the length of the remaining side. This is crucial when determining the possible side lengths of scalene and isosceles triangles.