# The Common Segment Theorem

In this post, we prove a theorem about a common segment between two segments. In the diagram below, $\overline{AC} \cong \overline{BD}$ and $\overline{BC}$ is a common segment to them. We show that if this is so,  $\overline{AB} \cong \overline{CD}$.

The Common Segment Theorem

Given: $\overline{AC} \cong \overline{BD}$.

Prove: $\overline{AB} = \overline {CD}$

Proof

1. $\overline{BC} \cong \overline{BC}$ by Reflexive Property. A segment is congruent to itself.

2a. $AC =AB + BC$ by Segment Addition PostulateThe Segment Addition Postulate states that if B is between $A$ and $C$, then $AC = AB + BC$.

2b. $BD = BC + CD$ by Segment Addition Postulate

3. $AB + BC = BC + CD$ Substitution.

4. $AB = CD$. Property of Equality (subtracting BC from both sides).

5. $\overline{AB} \cong \overline{CD}$ Definition of Congruent Segments.  Segments whose lengths are equal are congruent.

This proves the theorem.