Year 6 Maths reasoning (SATs): Classifying shapes, and proving or disproving
Part of the Year 6 Maths reasoning (SATs) geometry: angles and shapes chapter. Chapter overview · whole chapter on one page
Classifying shapes, and proving or disproving
Rule
A rectangle has four right angles and opposite sides equal. A square is a rectangle with four equal sides. A rhombus has four equal sides. A parallelogram has two pairs of parallel sides. A trapezium has one pair of parallel sides.
To disprove a statement, find one example where it is false.
Key ideas
- A square is a special rectangle and a special rhombus.
- An isosceles triangle has two equal sides, and an equilateral triangle has three.
- To disprove a statement, give one counter-example.
Method. A rhombus has four equal sides, and a square has four equal sides.
- Check the sides. A square has four equal sides.4 equal sides
- Compare with a rhombus. A rhombus has four equal sides, so every square is a rhombus.always
- Write the answer. always
Check. A square has four equal sides. ✓
Method. Find an isosceles triangle that is not equilateral.
- Pick a triangle. A triangle with sides 5, 5 and 7 has two equal sides.5, 5, 7
- Check it is not equilateral. Its sides are not all equal, so it is not equilateral.not equilateral
- Write the answer. no
Check. The triangle 5, 5, 7 is isosceles, not equilateral. ✓
Method. Know which quadrilateral has one pair of parallel sides.
- Parallelogram. A parallelogram has two pairs of parallel sides.two pairs
- Trapezium. A trapezium has exactly one pair of parallel sides.one pair
- Write the answer. trapezium
Check. One pair of parallel sides is a trapezium. ✓
Watch out
Watch out. A square is always a rhombus, but a rhombus is not always a square.
Exam tip
Practice guide. To disprove a statement, one counter-example is enough.
Exercise 5D Practice: classifying and proving
Non-calculatorMethod. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. A rhombus has four equal sides, but its angles are not all 90 degrees.rhombus fits
- Reject the other choices. A rectangle has four right angles but not equal sides. A square has four right angles.rectangle and square do not fit
- Write the answer. rhombus
Check. rhombus is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A rectangle has equal opposite sides and four right angles, and its diagonals cut each other in half.Opposite sides are equal, All angles are 90 degrees, The diagonals bisect each other
- Leave the others unticked. These do not match.All sides are equal are not ticked
- Write the answer. Opposite sides are equal, All angles are 90 degrees, The diagonals bisect each other
Check. Each ticked option is true. ✓
Watch out. All sides are equal only in a square, not in every rectangle.
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. The opposite sides of a parallelogram are equal.always fits
- Reject the other choices. It is not sometimes or never: the opposite sides are always equal.sometimes and never do not fit
- Write the answer. always
Check. always is the one that fits. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. Two right angles would already make 180 degrees, leaving nothing for the third angle.always fits
- Reject the other choices. A triangle cannot have two right angles, so it is not sometimes true.sometimes and never do not fit
- Write the answer. always
Check. always is the one that fits. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. Only an isosceles triangle with all three sides equal is equilateral.sometimes fits
- Reject the other choices. Not every isosceles triangle is equilateral, so it is not always true.always and never do not fit
- Write the answer. sometimes
Check. sometimes is the one that fits. ✓
Method. Compare the properties of a square and a rectangle.
- Properties of a rectangle. A rectangle has four right angles and opposite sides equal.four right angles
- Square. A square has four right angles and four equal sides, so it has the rectangle properties.yes
- Write the answer. yes
Check. A square has four right angles. ✓
Method. Find a triangle without a right angle.
- Pick a triangle. An equilateral triangle has angles of 60 degrees.60, 60, 60
- Check. None of the angles is 90 degrees, so it has no right angle.no right angle
- Write the answer. A triangle with angles 60, 60 and 60 has no right angle, so the statement is false.
Check. 60 + 60 + 60 = 180. ✓
Method. Think of a shape with four equal sides and angles that are not right angles.
- Pick a shape. A rhombus has four equal sides.four equal sides
- Check the angles. A rhombus can have angles of 60 and 120 degrees, so it is not a square.not 90 degrees
- Write the answer. Disproved, for example a rhombus with sides of 5 cm and angles of 60 and 120 degrees has four equal sides but is not a square.
Check. A rhombus with angles of 60 and 120 has four equal sides. ✓
Method. The three angles add up to 180 degrees.
- Add the expressions. 2x + 3x + 4x = 9x.9x
- Solve. 9x = 180, so x = 180 / 9 = 20.20
- Write the answer. 20
Check. 40 + 60 + 80 = 180. ✓
Method. Check which angle is bigger.
- Compare. 80 is bigger than 50.80
- Write the answer. 80
Check. 50 + 50 + 80 = 180. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. One angle is 90 degrees, and all three angles are different, so the sides are different.right-angled and scalene fits
- Reject the other choices. It is not equilateral, and it is not acute because of the 90 degree angle.equilateral and isosceles and acute do not fit
- Write the answer. right-angled and scalene
Check. right-angled and scalene is the one that fits. ✓
Method. Draw a parallelogram that leans over.
- Pick a parallelogram. A parallelogram with angles 60 and 120 degrees.60 and 120
- Check. Its angles are not all 90 degrees, so it is not a rectangle.not a rectangle
- Write the answer. No, because a parallelogram with angles 60 and 120 degrees is not a rectangle.
Check. 60 + 120 = 180, so the angles are not right angles. ✓
Method. Check which shape has two pairs of equal sides with the angles not all equal.
- Rectangle. A rectangle has equal opposite sides and all four angles equal (90 degrees).all angles equal
- Kite. A kite has two pairs of equal sides that are next to each other, and its angles are not all equal.kite
- Write the answer. kite
Check. The angles are not all equal, so it is not a rectangle. ✓