Complete the condition statements that must be met in order for three side lengths— a , b , and c —to create a triangle.

Question

a ?? b+c and a ?? b−c

Answer 1

To determine if three side lengths \( a \), \( b \), and \( c \) can form a triangle, the Triangle Inequality Theorem states that the following conditions must be satisfied:

1. \( a < b + c \) (the sum of the lengths of any two sides must be greater than the length of the remaining side)
2. \( b < a + c \)
3. \( c < a + b \)

Based on your request, the completed condition statements for side length \( a \) will be:

• \( a < b + c \)
• \( a > b - c \) (but it should also consider that \( b \) and \( c \) can be such that \( b - c \) could be negative; hence it's not a standalone condition in the context of triangle inequalities).

So, a more precise complete set would be:

1. \( a < b + c \)
2. \( a > |b - c| \) (to ensure \( a \) is greater than the absolute difference of \( b \) and \( c \))

However, in a general sense for triangles, it's often more succinct to consider the three conditions together to ensure that any length is valid in all contexts.