Let's work through the given information step-by-step and develop a clear solution:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
Step & Statement & Reason \\
\hline
1 & AC = AB + AB & Given \\
\hline
2 & AB + BC = AC & Segment Addition Postulate \\
\hline
3 & AB + BC = AB + AB & By substitution of AC from Step 1 in Step 2 \\
\hline
4 & BC = AB & Subtraction Property (Subtract AB from both sides of Step 3) \\
\hline
\end{tabular}
\][/tex]
After completing these steps, we can deduce the following:
1. We know from the given information that [tex]\( AC = AB + AB \)[/tex].
2. According to the Segment Addition Postulate, [tex]\( AB + BC = AC \)[/tex]. This means that the sum of the lengths of the segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] is equal to the length of [tex]\( AC \)[/tex].
3. By substituting [tex]\( AC \)[/tex] from Step 1 into Step 2, we get: [tex]\( AB + BC = AB + AB \)[/tex].
4. By subtracting [tex]\( AB \)[/tex] from both sides of the equation [tex]\( AB + BC = AB + AB \)[/tex], we find [tex]\( BC = AB \)[/tex].
So, we have two key conclusions:
- [tex]\( AB = BC \)[/tex] (derived using the Subtraction Property).
- From the above equality, if we substitute [tex]\( BC \)[/tex] with [tex]\( AB \)[/tex] in the initial statement [tex]\( AC = AB + AB \)[/tex], we maintain the relationship [tex]\( AC = AB + AB \)[/tex].
These relationships clarify:
- [tex]\( AB = BC \)[/tex] (segment equality).
- Since [tex]\( AC = AB + AB \)[/tex] can also be expressed as [tex]\( AC = 2AB \)[/tex] and we derived [tex]\( AB = BC \)[/tex]; hence, the length of [tex]\( AC \)[/tex] is twice the length of [tex]\( AB \)[/tex], and since [tex]\( AB = BC \)[/tex], the result [tex]\( AC = BC \)[/tex] holds naturally due to the equality [tex]\( AB = BC \)[/tex].
