To determine the valid length for side \( a \) in order to form a triangle with the other two sides, we can apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Given the possible responses (20, 13, 1, and 12), we need to know the lengths of sides \( b \) and \( c \) to apply these inequalities. However, without specific values for \( b \) and \( c \), we can analyze the choices based on possible reasonable scenarios using the Triangle Inequality conditions.
Let's consider these lengths in contexts:
- If \( b \) and \( c \) are 20 and 13, respectively:
- \( a + 13 > 20 \) → \( a > 7 \)
- \( a + 20 > 13 \) → \( a > -7 \) (always true)
- \( 13 + 20 > a \) → \( a < 33 \)
Thus \( 7 < a < 33 \). The choices that fit this range are 20 and 12.
- If \( b \) and \( c \) are, for example, 12 and 1:
- \( a + 1 > 12 \) → \( a > 11 \)
- \( a + 12 > 1 \) → \( a > -11 \) (always true)
- \( 1 + 12 > a \) → \( a < 13 \)
Thus \( 11 < a < 13 \) which would only allow \( a = 12 \).
From these analyses based on common values of \( b \) and \( c \), we find:
- If the existing sides allow \( a \) to take values of 12, 20, and perhaps, also 13, we can conclude:
- 12 is the likely candidate since it's usable in the tightest inequality allowing for smaller options.
Thus, the correct response for a that allows forming a valid triangle, given general scenarios, is 12.