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 \\(\overline{jn}\\), \\(\overline{jk}\\) and \\(\overline{lm}\\) marked congruent)
tabs: angles, segments, triangles, reasons
under tabs: \sas\ (reason: \given\), \def. of midpoint\ (reason: \corresponding angles theorem\)
statements | reasons
--- | ---
\\(\checkmark\\) 1. \\(\overline{jk} \parallel \overline{lm}\\) | 1. given
\\(\checkmark\\) 2. \\(\overline{jk} \cong \overline{lm}\\) | 2. given
\\(\triangle jlk \cong \triangle lnm\\) | blank
button: demo

Explanation:

Step1: Identify Corresponding Angles

Since \(\overline{JK} \parallel \overline{LM}\), by the Corresponding Angles Theorem, \(\angle JKL \cong \angle LMN\) (alternate interior angles, as \(JK \parallel LM\) and transversal \(KL\) or \(LM\) related? Wait, actually, the transversal here is \(JL\) or \(LN\)? Wait, no, let's correct: when \(JK \parallel LM\), and the transversal is \(KL\)? Wait, maybe better: \(\overline{JK} \parallel \overline{LM}\), so \(\angle KJL \cong \angle MLN\) (alternate interior angles, since \(JK \parallel LM\) and transversal \(JN\)). Wait, also, \(L\) is the midpoint of \(JN\), so \(JL = LN\) (definition of midpoint). We have \(JK \cong LM\) (given), \(\angle KJL \cong \angle MLN\) (alternate interior angles from parallel lines), and \(JL = LN\) (midpoint). So by SAS, \(\triangle JLK \cong \triangle LNM\).

Wait, let's structure the proof steps properly:

1. \(\overline{JK} \parallel \overline{LM}\) (given)
2. \(\overline{JK} \cong \overline{LM}\) (given)
3. \(L\) is the midpoint of \(\overline{JN}\) (given), so \(\overline{JL} \cong \overline{LN}\) (definition of midpoint)
4. \(\angle J \cong \angle MLN\) (alternate interior angles theorem, since \(JK \parallel LM\) and transversal \(JN\))
5. Therefore, \(\triangle JLK \cong \triangle LNM\) by SAS (side-angle-side: \(JL \cong LN\), \(\angle J \cong \angle MLN\), \(JK \cong LM\))

So the reason for the congruence is SAS (Side - Angle - Side) Congruence Theorem, using the sides \(JL \cong LN\), angle \(\angle J \cong \angle MLN\), and side \(JK \cong LM\).

Step2: Fill in the Reason

The last step in the proof (stating \(\triangle JLK \cong \triangle LNM\)) has the reason "SAS Congruence Theorem" because we have two sides and the included angle congruent: \(JL \cong LN\) (from midpoint), \(\angle J \cong \angle MLN\) (from parallel lines, alternate interior angles), and \(JK \cong LM\) (given).

Answer:

The reason for \(\triangle JLK \cong \triangle LNM\) is "SAS Congruence Theorem" (since we have two sides (\(\overline{JL} \cong \overline{LN}\), \(\overline{JK} \cong \overline{LM}\)) and the included angle (\(\angle J \cong \angle MLN\)) congruent, satisfying the SAS criteria).