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)
assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Identify given info

Given \(\overline{JK} \parallel \overline{LM}\), so alternate interior angles \(\angle LJK \cong \angle NLM\) (Alternate Interior Angles Theorem). Also, \(L\) is midpoint of \(\overline{JN}\), so \(\overline{JL} \cong \overline{LN}\) (Definition of Midpoint). And \(\overline{JK} \cong \overline{LM}\) (Given).

Step2: Apply SAS Congruence

In \(\triangle JLK\) and \(\triangle LNM\):

• \(\overline{JL} \cong \overline{LN}\) (from midpoint)
• \(\angle LJK \cong \angle NLM\) (alternate interior angles)
• \(\overline{JK} \cong \overline{LM}\) (given)

By SAS (Side - Angle - Side) Congruence Criterion, \(\triangle JLK \cong \triangle LNM\).

Answer:

To prove \(\triangle JLK \cong \triangle LNM\), we use the following steps:

Statements:

1. \(\overline{JK} \parallel \overline{LM}\), \(\overline{JK} \cong \overline{LM}\), \(L\) is the midpoint of \(\overline{JN}\) (Given)
2. \(\angle LJK \cong \angle NLM\) (Alternate Interior Angles Theorem, since \(\overline{JK} \parallel \overline{LM}\) and \(\overline{JL}\) is a transversal)
3. \(\overline{JL} \cong \overline{LN}\) (Definition of Midpoint, as \(L\) is the midpoint of \(\overline{JN}\))
4. \(\triangle JLK \cong \triangle LNM\) (SAS Congruence Criterion, using \(\overline{JL} \cong \overline{LN}\), \(\angle LJK \cong \angle NLM\), \(\overline{JK} \cong \overline{LM}\))

Reasons:

1. Given
2. Alternate Interior Angles Theorem
3. Definition of Midpoint
4. SAS Congruence Criterion