Question

given: ab = 12
ac = 6
prove: c is the midpoint of \\(\overline{ab}\\).

proof:
we are given that ab = 12 and ac = 6. applying the segment addition property, we get ac + cb = ab. applying the substitution property, we get 6 + cb = 12. the subtraction property can be used to find cb = 6. the symmetric property shows that 6 = ac. since cb = 6 and 6 = ac, ac = cb by the \\(\square\\) property. so, \\(\overline{ac} \cong \overline{cb}\\) by the definition of congruent segments. finally, c is the midpoint of \\(\overline{ab}\\) because it divides \\(\overline{ab}\\) into two congruent segments.

Explanation:

We know that \( CB = 6 \) and \( AC = 6 \) (from the symmetric property which shows \( 6 = AC \)). To conclude that \( AC = CB \), we use the transitive property of equality. The transitive property states that if \( a = b \) and \( b = c \), then \( a = c \). Here, let \( a = AC \), \( b = 6 \), and \( c = CB \). Since \( AC = 6 \) (from symmetric property) and \( 6 = CB \) (from the subtraction property result \( CB = 6 \)), by the transitive property, \( AC = CB \).

Answer:

transitive (of equality)