To determine which option shows a pair of angles that are not congruent when mapping triangle \( \triangle JKL \) onto triangle \( \triangle DEF \), we need to consider the properties of the angles in both triangles based on their respective positions in different quadrants.
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Triangle \( DEF \) is in quadrant II, which means:
- Angle \( D \) is an angle in that triangle.
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Triangle \( JKL \) is in quadrant IV, which means:
- Angle \( J \) is likely to be an angle in that triangle opposite \( D \).
Considering the placement of the triangles:
- Since \( \triangle DEF \) is in quadrant II, we know angle \( D \) is a second quadrant angle and thus is greater than \( 90^\circ \) but less than \( 180^\circ \).
- In contrast, triangle \( JKL \) is in quadrant IV, meaning angle \( J \) is less than \( 90^\circ \).
Given the typical properties of angles in these quadrants:
- Option #1: \( \angle L \cong \angle D \) - Correct if both angles are defined in a congruent way.
- Option #2: \( \angle K \cong \angle E \) - Again, correct if these angles are defined congruently.
- Option #3: \( \angle J \cong \angle D \) - This does not hold as angle \( J \) (Quadrant IV) is less than \( 90^\circ \) and angle \( D \) (Quadrant II) is greater than \( 90^\circ \), indicating these angles cannot be congruent.
- Option #4: \( \angle L \cong \angle F \) - This could be correct based on definition.
Therefore, Option #3 shows a pair of angles that are not congruent.
So the answer is:
Option #3 shows a pair of angles that are not congruent.