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}.

common segment theorem

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.

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