Question

using the sas congruence theorem
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 with vertices j, k, l, m, n; l on segment jn
statements | reasons
--- | ---
\checkmark 1. $overline{jk} \parallel \overline{lm}$ | 1. given
\checkmark 2. $overline{jk} \cong \overline{lm}$ | 2. given

1. l is the midpoint of $overline{jn}$. | 3. given

empty boxes for additional statements/reasons
correct! assemble the next statement.

Explanation:

Step1: Identify Alternate Interior Angles

Since \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JL}\) (or \(\overline{LN}\)) is a transversal, \(\angle JKL \cong \angle MLN\) (Alternate Interior Angles Theorem).

Step2: Use Midpoint Definition

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

Step3: Apply SAS Congruence

We have \(\overline{JK} \cong \overline{LM}\) (given), \(\angle JKL \cong \angle MLN\) (Alternate Interior Angles), and \(\overline{JL} \cong \overline{LN}\) (midpoint). So by SAS Congruence Theorem, \(\triangle JLK \cong \triangle LNM\).

For the next statement after step 3 (to build the proof table):

Step4: State \(\overline{JL} \cong \overline{LN}\)

From the midpoint definition (since \(L\) is midpoint of \(JN\), \(JL = LN\), so \(\overline{JL} \cong \overline{LN}\)).

Step5: State \(\angle JKL \cong \angle MLN\)

From Alternate Interior Angles (as \(JK \parallel LM\) and transversal \(KL\) (or \(LN\)/\(JL\))).

Step6: Apply SAS

With \(\overline{JK} \cong \overline{LM}\), \(\angle JKL \cong \angle MLN\), \(\overline{JL} \cong \overline{LN}\), \(\triangle JLK \cong \triangle LNM\) by SAS.

But focusing on the next statement after the given 3:

First, after statement 3 ("\(L\) is the midpoint of \(\overline{JN}\)"), the next statement is "\(\overline{JL} \cong \overline{LN}\)" (reason: definition of midpoint). Then, since \(\overline{JK} \parallel \overline{LM}\), the alternate interior angles \(\angle J \cong \angle MLN\) (wait, actually, the transversal is \(JL\) (or \(LN\))? Wait, the lines \(JK\) and \(LM\) are parallel, and the transversal is \(KL\)? Wait, no, looking at the diagram, the triangles are \(\triangle JLK\) and \(\triangle LNM\). So the sides: \(JK\) and \(LM\) are congruent (given), \(JL\) and \(LN\) are congruent (midpoint), and the included angle between \(JK\) & \(JL\) and \(LM\) & \(LN\) should be congruent. The included angle: since \(JK \parallel LM\), \(\angle JKL\) and \(\angle MLN\) are alternate interior angles? Wait, maybe \(\angle J\) and \(\angle MLN\)? Wait, no, let's re-examine.

Wait, the triangles are \(\triangle JLK\) and \(\triangle LNM\). So the sides:

- \(JK \cong LM\) (given)
- \(JL \cong LN\) (midpoint)
- The included angle: \(\angle JLK\) and \(\angle LNM\)? No, wait, the included angle between \(JK\) and \(JL\) is \(\angle J\), and between \(LM\) and \(LN\) is \(\angle M\)? No, maybe the alternate interior angles from \(JK \parallel LM\) and transversal \(JN\)? Wait, \(JK \parallel LM\), so \(\angle J \cong \angle MLN\) (corresponding angles? Wait, \(JN\) is a transversal? Wait, \(L\) is on \(JN\), so \(JN\) is a straight line. So \(JK \parallel LM\), so \(\angle J\) and \(\angle MLN\) are alternate interior angles (since \(JK\) and \(LM\) are parallel, and \(JN\) is the transversal). So \(\angle J \cong \angle MLN\).

But back to the proof table. After statement 3:

Statement 4: \(\overline{JL} \cong \overline{LN}\) (Reason: Definition of Midpoint)

Statement 5: \(\angle J \cong \angle MLN\) (Reason: Alternate Interior Angles Theorem, since \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JN}\) is a transversal)

Statement 6: \(\triangle JLK \cong \triangle LNM\) (Reason: SAS Congruence Theorem, since \(\overline{JK} \cong \overline{LM}\), \(\angle J \cong \angle MLN\), \(\overline{JL} \cong \overline{LN}\))

But the problem is to assemble the next statement after the first three. So the next statement (statement 4) should be \(\overline{JL} \cong \overline{LN}\) (from midpoint), then statement 5: \(\angle J \cong \angle MLN\) (alternate interior angles), then statement 6: SAS.

But let's follow the given options (the tiles: \(\cong\), \(\parallel\), \(\overline{JK}\), \(\overline{JL}\), \(\overline{LM}\), \(\overline{LN}\)). Wait, the tiles are for the statements. So after statement 3 ("\(L\) is the midpoint of \(\overline{JN}\)"), the next statement is about \(\overline{JL} \cong \overline{LN}\), so using the tiles, we can select \(\overline{JL}\), \(\cong\), \(\overline{LN}\) to form \(\overline{JL} \cong \overline{LN}\).

Answer:

The next statement (statement 4) is \(\boldsymbol{\overline{JL} \cong \overline{LN}}\) (reason: definition of midpoint). Then, the subsequent steps use alternate interior angles and SAS to prove the triangles congruent.