Let's start with examining the given statements, the statement that we want to prove and the figure.

Given : \overline{L K} ≅ \overline{N M}, \overline{K J} ≅ \overline{M J} Prove : \overline{L J} ≅ \overline{N J}

The given incomplete proof is shown below.

Statements	Reasons
$a .$ $\overline{L K} ≅ \overline{N M}, \overline{K J} ≅ \overline{M J}$	$a .$ $\underline{?}$
$b .$ $\underline{?}$	$b .$ Definition of Congruent Segments
$c .$ $L K + K J = N M + M J$	$c .$ $\underline{?}$
$d .$ $\underline{?}$	$d .$ Segment Addition Postulate
$e .$ $L J = N J$	$e .$ $\underline{?}$
$f .$ $\overline{L J} ≅ \overline{N J}$	$f .$ $\underline{?}$

We need to remember that to start a proof, we always state the given information first. Therefore, we can immediately fill in blank a since the corresponding statement is given.

\underline{a . Given}

For blank b, we will use the Definition of Congruent Segments. The definition says that two segments are congruent if and only if they have the same length.

\underline{b . L K = N M, K J = M J}

In the third step, the sum of the lengths of the segments is given. Since the numerical values are added, we can use the Addition Property of Equality to fill in the blank.

\underline{c . Addition Property of Equality}

For this part, we will consider the Segment Addition Postulate. The postulate says the following.

If A, B, and C are collinear, then point B is between A and C if and only if A B + B C = A C .

In this case, we can fill in blank d using the previous statement and looking at the given figure.

\underline{d . L J = L K + K J, N J = N M + M J}

In step five,

L J = N J

has been substituted for

L K + K J = N M + M J .

Therefore, we can use the Substitution Property of Equality.

\underline{e . Substitution Property of Equality}

As a last step, we can use the Definition of Congruent Segments that we stated in the second step.

\underline{f . Definition of Congruent Segments}

Combining these steps, we complete the two-column proof.

Statements	Reasons
$a .$ $\overline{L K} ≅ \overline{N M}, \overline{K J} ≅ \overline{M J}$	$a .$ Given
$b .$ $L K = N M, K J = K M$	$b .$ Definition of Congruent Segments
$c .$ $L K + K J = N M + M J$	$c .$ Addition Property of Equality
$d .$ $L J = L K + K J, N J = N M + M J$	$d .$ Segment Addition Postulate
$e .$ $L J = N J$	$e .$ Substitution Property of Equality
$f .$ $\overline{L J} ≅ \overline{N J}$	$f .$ Definition of Congruent Segments

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