Question

four parallel lines are drawn in a rectangular sign. the sign is painted using two colors in an alternating pattern, as shown.

which statements must be true about the two congruent, shaded figures? select two options.

• the figures are rectangles because they are formed inside a rectangle.
• the figures are rhombi because they are formed by four congruent line segments.
• the figures are trapezoids because each has exactly one pair of parallel sides.
• the figures are parallelograms because their opposite sides are parallel.
• the figures are quadrilaterals because each has four sides.

Explanation:

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<reasoning>

Analyze the geometric properties of the shaded figures

• The outer boundary is a rectangular sign, so the left and right vertical edges are parallel.
• The top and bottom boundaries of each shaded figure are formed by the drawn parallel lines.
• Thus, each shaded figure has two pairs of parallel opposite sides:

1. Left and right vertical sides (lying on the parallel vertical edges of the rectangle).
2. Top and bottom slanted sides (lying on the drawn parallel lines).

Evaluate the given statements

• Rectangles: The angles are not necessarily \(90^\circ\), so they are not guaranteed to be rectangles.
• Rhombi: The four sides are not guaranteed to be congruent.
• Trapezoids: They have two pairs of parallel sides, not exactly one pair.
• Parallelograms: True, because both pairs of opposite sides are parallel.
• Quadrilaterals: True, because each figure is a closed polygon with exactly four sides.

</reasoning>

<answer>
<mcq-option>The figures are rectangles because they are formed inside a rectangle.</mcq-option>
<mcq-option>The figures are rhombi because they are formed by four congruent line segments.</mcq-option>
<mcq-option>The figures are trapezoids because each has exactly one pair of parallel sides.</mcq-option>
<mcq-correct>The figures are parallelograms because their opposite sides are parallel.</mcq-correct>
<mcq-correct>The figures are quadrilaterals because each has four sides.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Analyze the geometric properties of the shaded figures

• The outer boundary is a rectangular sign, so the left and right vertical edges are parallel.
• The top and bottom boundaries of each shaded figure are formed by the drawn parallel lines.
• Thus, each shaded figure has two pairs of parallel opposite sides:

1. Left and right vertical sides (lying on the parallel vertical edges of the rectangle).
2. Top and bottom slanted sides (lying on the drawn parallel lines).

Evaluate the given statements

• Rectangles: The angles are not necessarily \(90^\circ\), so they are not guaranteed to be rectangles.
• Rhombi: The four sides are not guaranteed to be congruent.
• Trapezoids: They have two pairs of parallel sides, not exactly one pair.
• Parallelograms: True, because both pairs of opposite sides are parallel.
• Quadrilaterals: True, because each figure is a closed polygon with exactly four sides.

</reasoning>

<answer>
<mcq-option>The figures are rectangles because they are formed inside a rectangle.</mcq-option>
<mcq-option>The figures are rhombi because they are formed by four congruent line segments.</mcq-option>
<mcq-option>The figures are trapezoids because each has exactly one pair of parallel sides.</mcq-option>
<mcq-correct>The figures are parallelograms because their opposite sides are parallel.</mcq-correct>
<mcq-correct>The figures are quadrilaterals because each has four sides.</mcq-correct>
</answer>

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