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

\overline{A C}

and

C

is the midpoint of

\overline{B D} .

This is how we will begin our proof!

Statement 1) Reason 1) B is the midpoint of \overline{A C} C is the midpoint of \overline{B D} Given

By its own definition, a midpoint divides a segment in two congruent segments. Therefore, since

B

is the midpoint of

\overline{A C},

we know that

\overline{A B}

and

\overline{B C}

are congruent. Similarly, because

C

is the midpoint of

\overline{B D},

we know that

\overline{B C}

and

\overline{C D}

are also congruent.

Statement 2) Reason 2) \overline{A B} ≅ \overline{B C}, \overline{B C} ≅ \overline{C D} Definition of midpoint

Let's show this on the diagram.

By the definition of congruent segments, since

\overline{A B} ≅ \overline{B C},

we know that

A B = B C .

For the same reason, since

\overline{B C} ≅ \overline{C D},

we also know that

B C = C D .

Statement 3) Reason 3) A B = B C, B C = C D Definition of congruent segments

Knowing this, we can use the Transitive Property of Equality to show that

A B = C D .

{A B = B C B C = C D \Rightarrow A B = C D

We can list this step as the last step of our proof.

Statement 4) Reason 4) A B = C D Transitive Property of Equality

Completed Proof

Finally, we can complete our two-column table!

Statement	Reason
$B$ is the midpoint of $\overline{A C}$ $C$ is the midpoint of $\overline{B D}$	Given
$\overline{A B} ≅ \overline{B C}, \overline{B C} ≅ \overline{C D}$	Definition of midpoint
$A B = B C, B C = C D$	Definition of congruent segments
$A B = C D$	Transitive Property of Equality

Exercise