Let's begin by reviewing the idea of a two-column proof. It lists the statements on the left column and their corresponding justifications on the right column. Each statement must follow logically from its previous steps. In this case, we are given that B is the midpoint of AC and C is the midpoint of BD. This is how we will begin our proof!
Statement1)& B is the midpoint of AC
& C is the midpoint of BD
Reason1)& Given
By its own definition, a midpoint divides a segment in two congruent segments. Therefore, since B is the midpoint of AC, we know that AB and BC are congruent. Similarly, because C is the midpoint of BD, we know that BC and CD are also congruent.
Statement2)& AB≅ BC, BC≅ CD
Reason2)& Definition of midpoint
Let's show this on the diagram.
By the definition of congruent segments, since AB≅ BC, we know that AB= BC. For the same reason, since BC≅ CD, we also know that BC= CD.
Statement3)& AB=BC, BC=CD
Reason3)& Definition of congruent segments
Knowing this, we can use the Transitive Property of Equality to show that AB= CD.
AB = BC BC = CD ⇒ AB= CD
We can list this step as the last step of our proof.
Statement4)& AB=CD
Reason4)& Transitive Property of Equality
