Question

given: $overline{jk} \parallel overline{lm}$, $overline{jk} \cong overline{lm}$, l is the midpoint of $overline{jn}$. prove $\triangle jlk \cong \triangle lnm$

(diagram of triangles jlk and lnm with l on jn, jk || lm, jk ≅ lm)

statements | reasons
--- | ---

1. l is the midpoint of $overline{jn}$ | 1. given
2. $overline{jk} \cong overline{lm}$ | 2. given
3. $overline{jk} \parallel overline{lm}$ | 3. given

(empty boxes for next statement and reason)

prompt: correct! assemble the next statement.

Explanation:

Step1: Use midpoint definition

Since L is the midpoint of \(\overline{JN}\), by the definition of midpoint, \(\overline{JL} \cong \overline{LN}\).

Step2: Identify alternate interior angles

Because \(\overline{JK} \parallel \overline{LM}\), the alternate interior angles \(\angle J \cong \angle NLM\) (alternate interior angles theorem) and \(\angle K \cong \angle M\) (alternate interior angles theorem). But first, let's get the segment congruence from midpoint. Also, we have \(\overline{JK} \cong \overline{LM}\) (given). Now, for the triangles \(\triangle JLK\) and \(\triangle LNM\), we can use SAS (Side - Angle - Side) congruence. First, let's state the segment from midpoint: \(\overline{JL} \cong \overline{LN}\) (reason: definition of midpoint). Then, the angle between them: \(\angle J \cong \angle NLM\) (reason: alternate interior angles theorem, since \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JN}\) is a transversal). And we have \(\overline{JK} \cong \overline{LM}\) (given). But first, the next statement after the given ones should be about the midpoint giving \(\overline{JL} \cong \overline{LN}\). So the next statement is \(\overline{JL} \cong \overline{LN}\) with reason "definition of midpoint", then \(\angle J \cong \angle NLM\) (alternate interior angles) and then we can conclude congruence. But for the next statement after the three given, let's focus on the midpoint.

So the next statement (statement 4) should be: \(\overline{JL} \cong \overline{LN}\) (reason: definition of midpoint), then statement 5: \(\angle J \cong \angle NLM\) (reason: alternate interior angles theorem, as \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JN}\) is a transversal), then statement 6: \(\triangle JLK \cong \triangle LNM\) (reason: SAS congruence postulate, since \(\overline{JL} \cong \overline{LN}\), \(\angle J \cong \angle NLM\), \(\overline{JK} \cong \overline{LM}\)).

But let's build step by step. After the three given statements:

Step4: State segment from midpoint

Statement 4: \(\overline{JL} \cong \overline{LN}\)
Reason 4: definition of midpoint (since L is the midpoint of \(\overline{JN}\), it divides \(\overline{JN}\) into two congruent segments)

Step5: Identify alternate interior angle

Statement 5: \(\angle J \cong \angle NLM\)
Reason 5: alternate interior angles theorem (because \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JN}\) is a transversal, so the alternate interior angles are congruent)

Step6: Apply SAS congruence

Statement 6: \(\triangle JLK \cong \triangle LNM\)
Reason 6: SAS (Side - Angle - Side) congruence postulate (we have \(\overline{JL} \cong \overline{LN}\), \(\angle J \cong \angle NLM\), \(\overline{JK} \cong \overline{LM}\))

But if we are to do the next statement after the three given, the first next one is about the midpoint giving \(\overline{JL} \cong \overline{LN}\).

Answer:

The next statement (statement 4) is \(\boldsymbol{\overline{JL} \cong \overline{LN}}\) with reason "definition of midpoint". Then statement 5: \(\boldsymbol{\angle J \cong \angle NLM}\) (reason: alternate interior angles theorem), and statement 6: \(\boldsymbol{\triangle JLK \cong \triangle LNM}\) (reason: SAS congruence postulate).