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
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
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.