Question

Given: AB = 12
AC = 6
Prove: C is the midpoint of AB.

A line has points A, C, B.
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
property. So, AC ≅ CB by the definition of congruent segments. Finally, C is the midpoint of AB because it divides AB into two congruent segments.

Answer 1

The missing property is the transitive property of equality.

A concise proof:

1. Since A, C, B are collinear with C between A and B, by the Segment Addition Postulate AC + CB = AB.
2. Given AB = 12 and AC = 6, substitute to get 6 + CB = 12. (Substitution)
3. Subtract 6 from both sides to obtain CB = 6. (Subtraction property of equality)
4. From AC = 6 we have 6 = AC. (Symmetric property of equality)
5. From CB = 6 and 6 = AC, by the Transitive Property of equality, CB = AC.
6. Thus AC ≅ CB, so C divides AB into two congruent segments; hence C is the midpoint of AB (definition of midpoint).