In this post, we prove a theorem about a common segment between two segments. In the diagram below, and is a common segment to them. We show that if this is so, .

**The Common Segment Theorem**

Given: .

Prove:

**Proof**

1. by Reflexive Property. A segment is congruent to itself.

2a. by **Segment Addition Postulate. **The Segment Addition Postulate states that if B is between and , then .

2b. by Segment Addition Postulate

3. Substitution.

4. . Property of Equality (subtracting BC from both sides).

5. Definition of Congruent Segments. Segments whose lengths are equal are congruent.

This proves the theorem.