Year 7 Mathematics · AC9M7SP02

Classifying Polygons by Properties

We are learning to classify triangles, quadrilaterals and other polygons from defining side and angle properties

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Learning goalsSay it simply

We are learning to classify triangles, quadrilaterals and other polygons from defining side and angle properties.

A polygon belongs to a class because it satisfies the class's defining properties, not because it resembles a familiar prototype. Rotating, enlarging or reflecting a shape does not change its side lengths, angle measures, parallel relationships or class membership.

Inclusive definitions create a hierarchy of quadrilaterals. Every square is a rectangle because it has four right angles, every square is a rhombus because it has four equal sides, and both rectangles and rhombuses are parallelograms because they have two pairs of parallel opposite sides.

Triangle classifications can overlap because side properties and angle properties answer different questions. A triangle with two equal sides and one right angle is both isosceles and right-angled, while a regular polygon must have all sides equal and all interior angles equal.

Success criteria

  • I can use inclusive definitions to place a shape in every class it satisfies.
  • I can classify triangles by both side properties and angle properties.
  • I can justify a classification with necessary properties instead of relying on appearance.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Classifying a square through an inclusive hierarchy.

  1. Begin with a quadrilateral that has 4 equal sides and 4 right angles of 90 degrees.
  2. The 4 right angles satisfy the definition of a rectangle, while the 4 equal sides satisfy the definition of a rhombus.
  3. Both a rectangle and a rhombus have two pairs of parallel opposite sides, so the shape is also a parallelogram.
  4. List every valid class from most specific to broadest: square, rectangle, rhombus, parallelogram and quadrilateral.

Final answer: The shape is simultaneously a square, rectangle, rhombus, parallelogram and quadrilateral.

Check: Testing the defining properties of each listed class confirms every inclusion without using the shape's orientation or apparent proportions.

Example 2

Classifying a triangle by sides and angles.

  1. A triangle has angle measures 45 degrees, 45 degrees and 90 degrees, and the sides opposite the two 45-degree angles are equal.
  2. The two equal sides satisfy the isosceles definition.
  3. The 90-degree angle satisfies the right-angled triangle definition.
  4. The angle sum is 45 plus 45 plus 90, which equals 180 degrees and confirms that the measurements are possible for a triangle.

Final answer: The triangle is both isosceles and right-angled.

Check: The classification is supported independently by the equal-side condition and the one-right-angle condition.

Example 3

Application problem 1

Problem: Classify a quadrilateral with four equal sides and four right angles using every valid class name, and explain why giving only 'square' is incomplete.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: It is a square, rectangle, rhombus, parallelogram and quadrilateral. Inclusive classification means a shape belongs to every broader class whose defining properties it satisfies.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: It is a square, rectangle, rhombus, parallelogram and quadrilateral. Inclusive classification means a shape belongs to every broader class whose defining properties it satisfies.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: A quadrilateral has two pairs of parallel opposite sides and one right angle; prove the most specific class that is guaranteed.

  1. Plan: Use the consequences of opposite sides being parallel before deciding whether equal side lengths are known.
  2. Work: The shape is a rectangle because parallel opposite sides make adjacent interior angles supplementary, so the right angle forces all four angles to be 90 degrees; equal sides are not given, so a square is not guaranteed.
  3. Interpret: Parallelogram angle relationships extend one right angle to four right angles.

Final answer: The shape is a rectangle because parallel opposite sides make adjacent interior angles supplementary, so the right angle forces all four angles to be 90 degrees; equal sides are not given, so a square is not guaranteed.

Check: Parallelogram angle relationships extend one right angle to four right angles.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7SP02 - Classifying Polygons by Properties
Example 1

Example 1 Classifying a square through an inclusive hierarchy. Begin with a quadrilateral that has 4 equal sides and 4 right angles of 90 degrees. The 4 right angles satisfy the definition of a rectangle, while the 4 equal sides satisfy the definition of a rhombus. Both a rectangle and a rhombus have two pairs of parallel opposite sides, so the shape is also a parallelogram. List every valid class from most specific to broadest: square, rectangle, rhombus, parallelogram and quadrilateral. Final answer: The shape is simultaneously a square, rectangle, rhombus, parallelogram and quadrilateral. Check: Testing the defining properties of each listed class confirms every inclusion without using the shape's orientation or apparent proportions.

Example 2

Example 2 Classifying a triangle by sides and angles. A triangle has angle measures 45 degrees, 45 degrees and 90 degrees, and the sides opposite the two 45-degree angles are equal. The two equal sides satisfy the isosceles definition. The 90-degree angle satisfies the right-angled triangle definition. The angle sum is 45 plus 45 plus 90, which equals 180 degrees and confirms that the measurements are possible for a triangle. Final answer: The triangle is both isosceles and right-angled. Check: The classification is supported independently by the equal-side condition and the one-right-angle condition.

Example 3

Example 3 Application problem 1 Problem: Classify a quadrilateral with four equal sides and four right angles using every valid class name, and explain why giving only 'square' is incomplete. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: It is a square, rectangle, rhombus, parallelogram and quadrilateral. Inclusive classification means a shape belongs to every broader class whose defining properties it satisfies. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: It is a square, rectangle, rhombus, parallelogram and quadrilateral. Inclusive classification means a shape belongs to every broader class whose defining properties it satisfies. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: A quadrilateral has two pairs of parallel opposite sides and one right angle; prove the most specific class that is guaranteed. Plan: Use the consequences of opposite sides being parallel before deciding whether equal side lengths are known. Work: The shape is a rectangle because parallel opposite sides make adjacent interior angles supplementary, so the right angle forces all four angles to be 90 degrees; equal sides are not given, so a square is not guaranteed. Interpret: Parallelogram angle relationships extend one right angle to four right angles. Final answer: The shape is a rectangle because parallel opposite sides make adjacent interior angles supplementary, so the right angle forces all four angles to be 90 degrees; equal sides are not given, so a square is not guaranteed. Check: Parallelogram angle relationships extend one right angle to four right angles.

Curriculum examplesCopied content

Content description: classify triangles, quadrilaterals and other polygons according to their side and angle properties; identify and reason about relationships.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. A quadrilateral has four right angles, but its adjacent sides have different lengths; which most specific class is guaranteed?

    Check answer

    Answer: The quadrilateral is a rectangle.

    Hint: Classify from the angle information before considering the side lengths.

    Why: Four right angles are the defining property of a rectangle, while equal adjacent sides are not required.

  2. 2. A triangle has side lengths 6 centimetres, 6 centimetres and 9 centimetres; how is it classified by its sides?

    Check answer

    Answer: The triangle is isosceles because it has at least two equal sides.

    Hint: Compare the three given lengths and count how many are equal.

    Why: An isosceles triangle is identified by two or more equal side lengths.

  3. 3. State the two conditions that make a polygon regular.

    Check answer

    Answer: A regular polygon has all sides equal and all interior angles equal.

    Hint: Check one condition about side lengths and one condition about angle measures.

    Why: Regularity requires equality of both sides and angles.

  4. 4. A quadrilateral has four equal sides but no right angles; list every class that is guaranteed under the inclusive hierarchy.

    Check answer

    Answer: The shape is a rhombus, parallelogram and quadrilateral, but it is not guaranteed to be a rectangle or square.

    Hint: Move from the equal-side definition to broader classes and do not invent right angles.

    Why: Four equal sides guarantee a rhombus, whose opposite sides are parallel, without guaranteeing right angles.

  5. 5. A triangle has angles 30 degrees, 60 degrees and 90 degrees and three different side lengths; classify it by angles and by sides.

    Check answer

    Answer: The triangle is right-angled and scalene.

    Hint: Give one name from the angle information and one name from the side information.

    Why: One 90-degree angle makes it right-angled, and three different side lengths make it scalene.

  6. 6. A student says that a rotated square is a rhombus but no longer a rectangle; explain and correct the error.

    Check answer

    Answer: Rotation preserves the four right angles and four equal sides, so the shape remains a square, rectangle and rhombus.

    Hint: Check whether turning the shape changes any side length or angle measure.

    Why: Orientation is not a defining property and rigid rotation preserves lengths and angles.

  7. 7. A pentagon has five equal sides but two of its interior angles differ; is it regular, and why?

    Check answer

    Answer: It is not regular because a regular polygon requires all interior angles as well as all sides to be equal.

    Hint: Test both parts of the definition of a regular polygon.

    Why: Equal side lengths alone are insufficient to prove that a polygon is regular.

  8. 8. A quadrilateral has two pairs of parallel opposite sides and one right angle; prove the most specific class that is guaranteed.

    Check answer

    Answer: The shape is a rectangle because parallel opposite sides make adjacent interior angles supplementary, so the right angle forces all four angles to be 90 degrees; equal sides are not given, so a square is not guaranteed.

    Hint: Use the consequences of opposite sides being parallel before deciding whether equal side lengths are known.

    Why: Parallelogram angle relationships extend one right angle to four right angles.

  9. 9. Create the shortest yes-or-no decision sequence that distinguishes a general parallelogram, rectangle, rhombus and square once two pairs of opposite sides are known to be parallel.

    Check answer

    Answer: Ask whether all angles are right angles and whether all sides are equal; no-no gives a general parallelogram, yes-no gives a rectangle, no-yes gives a rhombus, and yes-yes gives a square.

    Hint: Use one question about angles and one question about side lengths, then consider all four answer combinations.

    Why: Two independent defining-property questions separate the four possible subclasses efficiently.

  10. 10. Classify a quadrilateral with four equal sides and four right angles using every valid class name, and explain why giving only 'square' is incomplete.

    Check answer

    Answer: It is a square, rectangle, rhombus, parallelogram and quadrilateral. Inclusive classification means a shape belongs to every broader class whose defining properties it satisfies.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: A square is not a rectangle because rectangles are longer than they are wide.

Correction: A rectangle is defined by four right angles, so a square is a special rectangle.

Common mistake: Turning a diamond-shaped quadrilateral changes whether it is a square or rhombus.

Correction: Orientation is not a defining property, so classification depends on marked sides, angles and parallel lines.

Common mistake: An isosceles triangle must have exactly two equal sides.

Correction: Under an inclusive definition, at least two equal sides are required, so an equilateral triangle also satisfies the isosceles condition.

Curriculum alignmentStart here

We are learning to classify triangles, quadrilaterals and other polygons from defining side and angle properties.

A polygon belongs to a class because it satisfies the class's defining properties, not because it resembles a familiar prototype. Rotating, enlarging or reflecting a shape does not change its side lengths, angle measures, parallel relationships or class membership.

Inclusive definitions create a hierarchy of quadrilaterals. Every square is a rectangle because it has four right angles, every square is a rhombus because it has four equal sides, and both rectangles and rhombuses are parallelograms because they have two pairs of parallel opposite sides.

Triangle classifications can overlap because side properties and angle properties answer different questions. A triangle with two equal sides and one right angle is both isosceles and right-angled, while a regular polygon must have all sides equal and all interior angles equal.

Success criteria

  • I can use inclusive definitions to place a shape in every class it satisfies.
  • I can classify triangles by both side properties and angle properties.
  • I can justify a classification with necessary properties instead of relying on appearance.
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