To prove something using segment addition, we typically use the Segment Addition Postulate. This postulate states that if a point $B$ lies on a line segment $AC$, then the sum of the lengths of segments $AB$ and $BC$ is equal to the length of $AC$.

Statement of Segment Addition Postulate:

$AB + BC = AC$

Example Proof Using Segment Addition:

Problem: Prove that if point $B$ lies between points $A$ and $C$ on a straight line, then $AB + BC = AC$.

Given:

• Points $A$, $B$, and $C$ are collinear.
• $B$ lies between $A$ and $C$.

To Prove: $AB + BC = AC$

Proof:

1. Start by identifying the segments:
   We have a line segment $AC$ with point $B$ between $A$ and $C$. Therefore, segments $AB$ and $BC$ exist, and $AC$ is the entire segment from $A$ to $C$.

2. Apply the Segment Addition Postulate:
   According to the Segment Addition Postulate, if a point $B$ lies on a line segment $AC$, then: $AB + BC = AC$

3. Conclude the proof:
   Since the conditions of the Segment Addition Postulate are satisfied (i.e., point $B$ lies between points $A$ and $C$), we have: $AB + BC = AC$

Therefore, the statement is proven.

Would you like any more details or have any questions?

Here are 5 related questions:

1. How does the Segment Addition Postulate help in solving geometric proofs?
2. Can you provide an example where the Segment Addition Postulate is used in a coordinate plane?
3. What happens if point $B$ does not lie between $A$ and $C$?
4. How does the Segment Addition Postulate relate to the concept of betweenness?
5. Can the Segment Addition Postulate be applied to non-collinear points?

Tip: When solving problems involving segments, always check if you can apply the Segment Addition Postulate. This can simplify many geometric proofs.