To solve this problem, let’s analyze the transformations step by step and understand their effects on the properties of polygon $JKLM$:

1. Translation of 4 units left

• A translation is a rigid transformation, meaning it does not change the size, shape, or orientation of the polygon. The angle measures, side lengths, perimeter, and area remain unchanged.

2. Reflection across the $x$-axis

• A reflection is also a rigid transformation, preserving size, shape, and angle measures. The polygon is flipped over the $x$-axis, but this does not alter the side lengths, perimeter, or area.

Based on these transformations:

• Angle Measures: Reflections and translations do not affect angle measures, so the angle measures of $QRST$ are equal to the corresponding angle measures of $JKLM$. This makes Option A incorrect.
• Sum of Angle Measures: The sum of the angles in a polygon is dependent on the number of sides and does not change under rigid transformations. Thus, Option B is incorrect.
• Perimeter: Since the side lengths are unchanged, the perimeters of $JKLM$ and $QRST$ are equal. Option C is correct.
• Side Lengths: Reflections and translations do not affect side lengths, so the side lengths of $JKLM$ are congruent to those of $QRST$. Option D is correct.
• Area: Rigid transformations do not change the area, so the area of $QRST$ is equal to the area of $JKLM$. This makes Option E incorrect.

Correct Responses:

• C. The perimeter of polygon $QRST$ is congruent to the perimeter of polygon $JKLM$.
• D. The side lengths of polygon $QRST$ are congruent to the side lengths of polygon $JKLM$.

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Would you like a detailed explanation of any part of this solution? Here are 5 related questions to consider:

1. What are rigid transformations, and how do they affect geometric figures?
2. How do reflections across axes change the coordinates of a polygon’s vertices?
3. Why does a polygon’s perimeter remain unchanged after a reflection or translation?
4. Can rigid transformations ever alter the orientation of a figure?
5. What properties of a polygon are preserved under rigid transformations?

Tip: When dealing with transformations, always focus on whether they preserve the size, shape, and orientation of the figure.