2D Figure Classification

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Answer Key & Explanations

2D Figure Classification · Grade 5 · Worksheet 2

1. Kaia is helping her younger brother sort his shape puzzle pieces. She notices that some shapes can be grouped into a special category. Her brother holds up a rhombus and says, 'This shape has four equal sides, so it must be a square!' Kaia explains that while all squares are rhombuses, not all rhombuses are squares. Which of the following statements correctly explains the relationship between rhombuses and squares? Answer: C. A square is a special type of rhombus with four right angles, so every square is a rhombus Solution: A rhombus is defined as a quadrilateral with all four sides equal. A square is a quadrilateral with all four sides equal AND all four angles equal (each 90 degrees).
   Full step-by-step solution

   Step 1: A rhombus is defined as a quadrilateral with all four sides equal. Step 2: A square is a quadrilateral with all four sides equal AND all four angles equal (each 90 degrees). Step 3: Since a square has all sides equal, it meets the definition of a rhombus. Therefore, every square is a rhombus. Step 4: However, not all rhombuses have right angles (example: a diamond shape with equal sides but slanted). So not every rhombus is a square. Step 5: The correct statement is: 'A square is a special type of rhombus with four right angles, so every square is a rhombus.' The correct answer is A square is a special type of rhombus with four right angles, so every square is a rhombus.

2. Maria is designing a stained glass window using different geometric shapes. She needs a polygon with exactly 5 sides where all sides are equal and all interior angles are equal. What is the name of this polygon, and what is the sum of its interior angles? Answer: A. Regular Pentagon, 540° Solution: A polygon with exactly 5 sides is called a pentagon. When all sides are equal and all interior angles are equal, it is called a regular pentagon.
   Full step-by-step solution

   Step 1: A polygon with exactly 5 sides is called a pentagon. Step 2: When all sides are equal and all interior angles are equal, it is called a regular pentagon. Step 3: To find the sum of interior angles of any polygon, use the formula: (n - 2) × 180°, where n is the number of sides. Step 4: For a pentagon, n = 5, so the sum of interior angles = (5 - 2) × 180° = 3 × 180° = 540°. Step 5: Therefore, the polygon is a regular pentagon with interior angles summing to 540°. The correct answer is Regular Pentagon, 540°.

3. Mere has drawn a large quadrilateral on the pavement. It has two pairs of parallel sides, and all four sides are of equal length, but its angles are not 90 degrees. A friend says, 'That must be a square.' Is the friend correct? Explain your reasoning by naming the most specific shape that Mere has drawn and listing its properties. Answer: No, the friend is not correct. The shape is a rhombus. A rhombus is a quadrilateral with all four sides equal in length and two pairs of parallel sides (opposite sides are parallel). While a square also has these properties, a square must additionally have four right angles (90 degrees). Since Mere's shape does not have 90-degree angles, it is a rhombus but not a square. Solution: List the properties of the shape Mere drew: quadrilateral (4 sides), two pairs of parallel sides (parallelogram), all four sides equal length, angles are not 90 degrees. Check if this matches a square.
   Full step-by-step solution

   Step 1: List the properties of the shape Mere drew: quadrilateral (4 sides), two pairs of parallel sides (parallelogram), all four sides equal length, angles are not 90 degrees. Step 2: Check if this matches a square. A square must have: 4 equal sides, 2 pairs of parallel sides, AND 4 right angles (90 degrees each). Mere's shape does NOT have 90-degree angles, so it cannot be a square. Step 3: Identify the correct shape. A rhombus is a quadrilateral with all sides equal and opposite sides parallel. Angles do not have to be 90 degrees. Therefore, Mere's shape is a rhombus. Step 4: Conclusion: The friend is incorrect. The most specific name for Mere's shape is a rhombus.

4. Aroha is sorting shapes into groups based on their properties. She has four shapes: a square, a rhombus, a rectangle, and a trapezoid. She claims, "Every square is a rhombus, and every rhombus is a parallelogram, so every square must also be a parallelogram." Kaia disagrees and says, "A square is a rectangle, but a rectangle is not always a rhombus, so squares and rhombuses are not related that way." Who is correct about the relationships? Explain using the definitions of the shapes. Answer: Aroha is correct. A square has all sides equal (like a rhombus) and opposite sides parallel (like a parallelogram). Every square is a rhombus because all sides are equal, and every rhombus is a parallelogram because opposite sides are parallel. Therefore, every square is also a parallelogram. Kaia is partially correct that every square is a rectangle, but she incorrectly says squares and rhombuses are not related—they are related through the property of equal sides. Solution: Define a square: a quadrilateral with all sides equal and all angles 90 degrees. Step 2: Define a rhombus: a quadrilateral with all sides equal (angles can vary).
   Full step-by-step solution

   Step 1: Define a square: a quadrilateral with all sides equal and all angles 90 degrees. Step 2: Define a rhombus: a quadrilateral with all sides equal (angles can vary). Step 3: A square has all sides equal, so it meets the definition of a rhombus. Therefore, every square is a rhombus. Step 4: Define a parallelogram: a quadrilateral with both pairs of opposite sides parallel. Step 5: A rhombus has opposite sides parallel (because equal sides force parallel opposite sides in a quadrilateral). Therefore, every rhombus is a parallelogram. Step 6: Since every square is a rhombus, and every rhombus is a parallelogram, every square is also a parallelogram. Step 7: Kaia's claim that squares and rhombuses are not related is false because they share the property of equal sides. The correct answer is Aroha is correct.

5. Emma is helping her younger brother sort a set of plastic shapes for a school project. She finds a shape that has 4 sides, 4 right angles, and opposite sides that are parallel and equal in length. Her brother says it is a square, but Emma thinks it could also be classified as another type of quadrilateral. What is the broader category of quadrilateral that this shape definitely belongs to, and is every shape in that category also a square? Answer: Rectangle; No, not every rectangle is a square. Solution: Identify the shape's properties: 4 sides, 4 right angles, opposite sides parallel and equal. This describes both a square and a rectangle. A square is a special type of rectangle where all four sides are equal.
   Full step-by-step solution

   Step 1: Identify the shape's properties: 4 sides, 4 right angles, opposite sides parallel and equal. This describes both a square and a rectangle. Step 2: A square is a special type of rectangle where all four sides are equal. However, the broader category of 'rectangle' includes shapes where only opposite sides must be equal, not all four sides. Step 3: Therefore, the shape is definitely a rectangle. The answer to the second part is no: a rectangle does not have to have all sides equal, so not every rectangle is a square. The correct answer is: Rectangle; No, not every rectangle is a square.