2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 2

1. Mason is drawing a family tree of quadrilaterals for his geometry project. He says, 'Every square is a rectangle, but not every rectangle is a square.' Charlotte disagrees and says, 'No, every rectangle is a square, but not every square is a rectangle.' Isabella, their teacher, asks the class: Who is correct, and why? Explain using the properties of sides and angles. Answer: Mason is correct. Every square is a rectangle because a square has four right angles and opposite sides parallel and equal (all sides equal, which still meets the rectangle definition). But not every rectangle is a square because a rectangle can have different side lengths, while a square must have all four sides equal. Solution: In geometry, a rectangle is defined as a quadrilateral with four right angles and opposite sides parallel and equal. A square is a special type of rectangle that also has all four sides equal.
   Full step-by-step solution

   In geometry, a rectangle is defined as a quadrilateral with four right angles and opposite sides parallel and equal. A square is a special type of rectangle that also has all four sides equal. So every square automatically meets the rectangle definition, but rectangles with unequal side lengths are not squares. This is a hierarchical relationship — think of squares as a 'subset' of rectangles, like how all robins are birds but not all birds are robins.

2. Aroha is helping her younger brother sort his collection of pattern blocks for a school project. He has a pile of quadrilaterals that includes a square with sides of 12 cm, a rhombus with equal sides but no right angles, a rectangle that is 18 cm long and 9 cm wide, and a parallelogram that is not a rectangle. Aroha's brother claims that all four shapes are parallelograms. Is Aroha's brother correct? Explain your reasoning by identifying which shapes are parallelograms and which are not, using the properties of each shape. Answer: No, Aroha's brother is not correct. The square, rhombus, and rectangle are all parallelograms because they each have two pairs of parallel sides. However, the parallelogram that is 'not a rectangle' is also a parallelogram by definition. Wait—the problem states he has a parallelogram that is not a rectangle, so actually all four shapes are parallelograms. The correct answer is: Yes, all four shapes are parallelograms. The square is a parallelogram (all squares are parallelograms), the rhombus is a parallelogram (all rhombuses are parallelograms), the rectangle is a parallelogram (all rectangles are parallelograms), and the fourth shape is explicitly a parallelogram. So his brother is correct. Solution: Recall the definition of a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Check the square: A square has all sides equal and all angles 90 degrees. Its opposite sides are parallel.
   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 square: A square has all sides equal and all angles 90 degrees. Its opposite sides are parallel. Therefore, a square is a parallelogram. Step 3: Check the rhombus: A rhombus has all sides equal but angles are not necessarily 90 degrees. Its opposite sides are parallel. Therefore, a rhombus is a parallelogram. Step 4: Check the rectangle: A rectangle has opposite sides equal and all angles 90 degrees. Its opposite sides are parallel. Therefore, a rectangle is a parallelogram. Step 5: The fourth shape is explicitly stated as a parallelogram, so it is a parallelogram. Step 6: Since all four shapes satisfy the definition of having two pairs of parallel sides, Aroha's brother is correct. All squares, rectangles, and rhombuses are special types of parallelograms. Final answer: Yes, all four shapes are parallelograms.

3. Noah is helping his aunt design a new garden layout. She wants to plant flowers in a shape that has four sides, with both pairs of opposite sides parallel and all four sides equal in length, but the angles are not necessarily 90 degrees. Noah's aunt says, 'I know that this shape is also a special type of parallelogram.' Which statement correctly describes the relationship between this shape and other quadrilaterals? Answer: C. Every rhombus is a parallelogram, but not every parallelogram is a rhombus. Solution: Identify the shape Noah's aunt described: a quadrilateral with both pairs of opposite sides parallel and all four sides equal. This is a rhombus. By definition, a parallelogram has both pairs of opposite sides parallel.
   Full step-by-step solution

   Step 1: Identify the shape Noah's aunt described: a quadrilateral with both pairs of opposite sides parallel and all four sides equal. This is a rhombus. Step 2: By definition, a parallelogram has both pairs of opposite sides parallel. Since a rhombus meets this condition, every rhombus is a parallelogram. Step 3: However, a parallelogram only needs opposite sides parallel; it does not require all sides to be equal (e.g., a rectangle is a parallelogram with equal opposite sides, not all four sides equal). So not every parallelogram is a rhombus. Step 4: Check the other options: A square is a special rhombus with right angles, so option D is true but not the best description of the relationship between rhombus and parallelogram. Option A is false (not every rhombus has right angles). Option C is false (not every parallelogram has equal sides). Step 5: The correct statement is option B: Every rhombus is a parallelogram, but not every parallelogram is a rhombus.

4. Emma is sorting a set of quadrilaterals into a hierarchy chart. She has drawn a shape with four equal sides but its angles are not all 90 degrees. Which category does this shape belong to? Is it also a rectangle? Explain your reasoning using the properties of the shape. Answer: The shape is a rhombus. It is not a rectangle because rectangles must have four right angles, and this shape does not have 90-degree angles. Solution: The shape has four equal sides, so it could be a square or a rhombus. Step 2: Since the angles are not all 90 degrees, it cannot be a square or a rectangle.
   Full step-by-step solution

   Step 1: The shape has four equal sides, so it could be a square or a rhombus. Step 2: Since the angles are not all 90 degrees, it cannot be a square or a rectangle. Step 3: A rhombus is defined as a quadrilateral with all sides equal, which matches. Step 4: A rectangle requires all angles to be 90 degrees, which this shape does not have. Therefore, the shape is a rhombus but not a rectangle.