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).

Concise proof:
1. A, C, B are collinear with C between A and B, so AC + CB = AB. (Segment addition)
2. Given AC = 6, substitute into step 1: 6 + CB = AB. (Substitution)
3. Given AB = 12, 6 + CB = 12, so CB = 6. (Subtraction)
4. From AC = 6 we also have 6 = AC. (Symmetric property)
5. CB = 6 and 6 = AC imply CB = AC. (Transitive property of equality)
6. Hence AC ≅ CB (congruent segments). Since C lies between A and B and divides AB into two congruent segments, C is the midpoint of AB.