2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 1

1. Tane is helping his younger sister sort her geometric puzzle pieces. She has a shape that always has four sides and four right angles. Tane tells her, 'This shape is a rectangle, but it could also be called something more specific if all sides are equal.' Which of the following statements about the shape is always true? Answer: D. The shape must be a parallelogram Solution: Identify the properties given: four sides (quadrilateral) and four right angles. A shape with four right angles is a rectangle. A rectangle always has opposite sides parallel, which makes it a parallelogram.
   Full step-by-step solution

   Step 1: Identify the properties given: four sides (quadrilateral) and four right angles. Step 2: A shape with four right angles is a rectangle. Step 3: A rectangle always has opposite sides parallel, which makes it a parallelogram. Step 4: Not all rectangles are squares (only those with equal sides). Step 5: Not all rectangles are rhombuses (rhombuses need equal sides but not right angles). Step 6: A trapezoid has only one pair of parallel sides, but a rectangle has two pairs. Step 7: The only statement that is always true is that the shape must be a parallelogram. The correct answer is The shape must be a parallelogram.

2. Tane is designing a garden path using tiles that are all quadrilaterals. He has a tile with four right angles, but its sides are not all the same length. His friend Kaia says, 'That tile must be a rectangle, so it is also a parallelogram.' Tane wonders if Kaia is correct. Is every rectangle also a parallelogram? Explain your reasoning using the properties of these shapes. Answer: Yes, every rectangle is also a parallelogram because a rectangle has two pairs of parallel sides (opposite sides are parallel), which is the definition of a parallelogram. Solution: A parallelogram is any quadrilateral where both pairs of opposite sides are parallel.
   Full step-by-step solution

   A parallelogram is any quadrilateral where both pairs of opposite sides are parallel. A rectangle is a special type of parallelogram because its opposite sides are parallel, but it has the extra property of having four right angles. So, all rectangles are parallelograms, but not all parallelograms are rectangles.

3. Isabella is sorting a collection of geometric cutouts for a school art project. She has a square, a rectangle, a rhombus, and a parallelogram. Her friend Mason claims that a square can be classified as both a rhombus and a rectangle. Is Mason correct? Explain why or why not using the properties of these shapes. Answer: Yes, Mason is correct. A square has all sides equal (property of a rhombus) and all angles 90 degrees (property of a rectangle), so it fits the definitions of both a rhombus and a rectangle. Solution: List the properties of a square: 4 equal sides and 4 right angles. List the properties of a rhombus: 4 equal sides (but angles do not have to be right).
   Full step-by-step solution

   Step 1: List the properties of a square: 4 equal sides and 4 right angles. Step 2: List the properties of a rhombus: 4 equal sides (but angles do not have to be right). Step 3: List the properties of a rectangle: 4 right angles (but sides do not have to be equal). Step 4: Compare: A square has 4 equal sides (meets rhombus definition) and 4 right angles (meets rectangle definition). Step 5: Conclusion: Yes, a square is both a rhombus and a rectangle. Mason is correct. The answer is: Yes, Mason is correct.

4. Emma has drawn a quadrilateral with the following properties: it has exactly one pair of parallel sides, and it has no right angles. Liam says, 'That shape must be a trapezoid.' Olivia says, 'But it could also be a parallelogram because it has parallel sides.' Who is correct, and why? Explain using the hierarchical relationships between quadrilaterals. Answer: Emma is correct. The shape is a trapezoid because it has exactly one pair of parallel sides. Liam is incorrect because a parallelogram requires two pairs of parallel sides, and this shape only has one. In the hierarchy of quadrilaterals, all parallelograms are trapezoids (if using the inclusive definition), but not all trapezoids are parallelograms. Since this shape has only one pair of parallel sides, it can only be classified as a trapezoid, not a parallelogram. Solution: Quadrilaterals are classified by their side properties. A trapezoid (in the exclusive definition used in many schools) has exactly one pair of parallel sides. A parallelogram has two pairs of parallel sides.
   Full step-by-step solution

   Quadrilaterals are classified by their side properties. A trapezoid (in the exclusive definition used in many schools) has exactly one pair of parallel sides. A parallelogram has two pairs of parallel sides. Since the shape in the problem only has one pair of parallel sides, it fits the definition of a trapezoid but not a parallelogram. Remember: in a hierarchy, more specific shapes (like rectangles, rhombuses, squares) belong to broader categories (like parallelograms, trapezoids), but the reverse is not always true. For example, a square is always a rectangle, but a rectangle is not always a square.

5. Isabella is helping her art teacher organize a display of geometric shapes. She has a bin labeled 'Quadrilaterals' that contains a rhombus, a square, a trapezoid, and a rectangle. The teacher asks her to move any shape that is also a parallelogram into a new bin. Which shapes from the 'Quadrilaterals' bin should Isabella move into the 'Parallelogram' bin? Explain your reasoning. Answer: Square, rectangle, and rhombus Solution: Recall the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Check the rhombus: It has two pairs of parallel sides, so it IS a parallelogram.
   Full step-by-step solution

   Step 1: Recall the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Step 2: Check the rhombus: It has two pairs of parallel sides, so it IS a parallelogram. Step 3: Check the square: It has two pairs of parallel sides, so it IS a parallelogram. Step 4: Check the rectangle: It has two pairs of parallel sides, so it IS a parallelogram. Step 5: Check the trapezoid: By definition, a trapezoid has exactly one pair of parallel sides, so it is NOT a parallelogram. Step 6: Therefore, Isabella should move the rhombus, square, and rectangle into the 'Parallelogram' bin.

6. Charlotte is designing a pattern for a new playground. She needs to sort a set of quadrilaterals into groups based on their properties. She has a square, a rhombus, and a rectangle. She wants to know if every square can be called a rectangle. Is Charlotte correct? Explain why or why not using the properties of these shapes. Answer: Yes, every square is a rectangle because a square has all the properties of a rectangle: four right angles and opposite sides that are parallel and equal. The additional property of equal sides does not disqualify it from being a rectangle. Solution: Recall the properties of a rectangle: It is a quadrilateral with four right angles, and opposite sides are parallel and equal.
   Full step-by-step solution

   Step 1: Recall the properties of a rectangle: It is a quadrilateral with four right angles, and opposite sides are parallel and equal. Step 2: Recall the properties of a square: It is a quadrilateral with four right angles, all four sides equal, and opposite sides parallel. Step 3: Compare: A square has four right angles and opposite sides that are parallel and equal, just like a rectangle. The only difference is that a square has all sides equal, but that does not break any rectangle rules. Step 4: Therefore, every square meets all the conditions to be a rectangle, so Charlotte is correct. The answer is: Yes, every square is a rectangle.