GCSE Quadrilateral properties Practice Questions

Free GCSE Quadrilateral properties practice questions with full step-by-step worked solutions. Covers angle sum of a quadrilateral, forming an equation, parallelogram, co-interior angles. Practise exam-style problems and check your method.

angle sum of a quadrilateralforming an equationparallelogramco-interior anglesopposite anglesrhombus
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
The four angles of a quadrilateral ABCDABCD are 78∘78^\circ, 95∘95^\circ, 112∘112^\circ and x∘x^\circ. Work out the value of xx.
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Worked solution

  1. Use the angle sum of a quadrilateral

    78+95+112+x=36078 + 95 + 112 + x = 360

    The angles of any quadrilateral add up to 360°.

  2. Subtract the three known angles

    x=360−285=75x = 360 - 285 = 75

    The three known angles come to 285°, and 360 minus 285 is 75.

  3. State the answer

    x=75x = 75

    The fourth angle is 75°.

Answer
x=75x = 75
Question 2
2 markseasy
A quadrilateral has these properties. It has two pairs of equal adjacent sides, but its four sides are not all equal. Its diagonals cross at right angles. What is the most precise name for it?
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Worked solution

  1. Use the clue: It has two pairs of equal adjacent sides, but its four sides are not all equal

    AB=AD,CB=CD,AB≠CBAB = AD, \quad CB = CD, \quad AB \neq CB

    Equal sides that are next to each other, not opposite each other, is the defining property of a kite. Because the four sides are not all equal it is not a rhombus.

  2. Use the clue: Its diagonals cross at right angles

    AC⊥BDAC \perp BD

    Diagonals that cross at right angles happen in a square, a rhombus and a kite.

  3. State the name of the shape

    Kite\text{Kite}

    The quadrilateral described is a kite.

Answer
Kite\text{Kite}
Question 3
2 marksintermediate
The diagonals of a rhombus ABCDABCD meet at MM. Angle BAD=76∘BAD = 76^\circ. Work out the size of angle ABMABM.
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Worked solution

  1. Write down what the question tells you

    ∠BAD=76∘\angle BAD = 76^\circ

    List what you are given before you start, and make sure every piece of it is used somewhere.

  2. Recall the properties of a rhombus

    4 equal sides,opposite angles equal\text{4 equal sides},\quad \text{opposite angles equal}

    A rhombus has four equal sides, so it is a parallelogram: opposite sides are parallel and opposite angles are equal. Its diagonals bisect each other at right angles and bisect the corner angles. It has two lines of symmetry (its diagonals) and rotational symmetry of order 2.

  3. Use the diagonal that bisects the angle

    ∠MAB=76∘2=38∘\angle MAB = \frac{76^\circ}{2} = 38^\circ

    The diagonal AC bisects angle BAD, so the angle between AB and AM is half of 76°.

  4. Use the perpendicular diagonals

    ∠AMB=90∘\angle AMB = 90^\circ

    The diagonals of a rhombus cross at right angles, so angle AMB is a right angle.

  5. Use the angle sum of triangle ABM

    ∠ABM=180∘−90∘−38∘=52∘\angle ABM = 180^\circ - 90^\circ - 38^\circ = 52^\circ

    The three angles of triangle ABM add up to 180°.

  6. State the answer

    ∠ABM=52∘\angle ABM = 52^\circ

    Angle ABM measures 52°.

Answer
52∘52^\circ
Question 4
4 markshard
A parallelogram has diagonals that are equal in length. Which one of these statements about the parallelogram is true?
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Worked solution

  1. Split the parallelogram with its two diagonals

    AC=BD,OA=OB=OC=ODAC = BD, \quad OA = OB = OC = OD

    The diagonals of any parallelogram bisect each other, so the four half diagonals are all half of a diagonal. If the two diagonals are also equal, all four halves are equal.

  2. Look at the triangles the diagonals create

    OA=OB⇒∠OAB=∠OBAOA = OB \Rightarrow \angle OAB = \angle OBA

    Every one of the four triangles round the crossing point is isosceles, because two of its sides are half diagonals.

  3. Work out the angle at a corner

    ∠DAB=∠OAB+∠OAD=∠OBA+∠ODA\angle DAB = \angle OAB + \angle OAD = \angle OBA + \angle ODA

    The corner angle at A is made of one base angle from each of the two triangles that meet there, and those base angles are copied at B and at D.

  4. Deduce that the corner angles are right angles

    ∠DAB=90∘\angle DAB = 90^\circ

    Going round the parallelogram, the four corner angles are built from the same four base angles, so all four corners are equal. Four equal angles in a quadrilateral must each be 360/4=90∘360 / 4 = 90^\circ, so the shape is a rectangle.

  5. Rule out the first of the other statements

    counter-example 1\text{counter-example 1}

    "A parallelogram whose diagonals are equal in length must be a rhombus" is false. A 6 cm by 3 cm rectangle is a parallelogram with equal diagonals, and its sides are not all equal.

  6. Rule out the second of the other statements

    counter-example 2\text{counter-example 2}

    "A parallelogram whose diagonals are equal in length must be a square" is false. A 6 cm by 3 cm rectangle is a parallelogram with equal diagonals but it is not a square.

  7. Rule out the third of the other statements

    counter-example 3\text{counter-example 3}

    "A parallelogram whose diagonals are equal in length must be a kite" is false. A 6 cm by 3 cm rectangle has equal diagonals and its equal sides are opposite each other, not adjacent, so it is not a kite.

  8. Rule out the fourth of the other statements

    counter-example 4\text{counter-example 4}

    "A parallelogram whose diagonals are equal in length can have no right angles at all" is false. Equal diagonals in a parallelogram force all four angles to be 90°, so it must have right angles.

  9. Check that only one statement survives

    4 false, 1 true\text{4 false, 1 true}

    Each of the other four statements has a counter-example, so exactly one statement is left standing.

  10. State the true statement

    true\text{true}

    A parallelogram whose diagonals are equal in length must be a rectangle.

Answer
A parallelogram whose diagonals are equal in length must be a rectangle\text{A parallelogram whose diagonals are equal in length must be a rectangle}
Question 5
6 markschallenging
The vertices of a quadrilateral ABCDABCD are A(1,1)A(1, 1), B(6,1)B(6, 1), C(8,5)C(8, 5) and D(3,5)D(3, 5). What is the most precise name for the quadrilateral ABCDABCD?
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Worked solution

  1. Work out the vector along each side

    AB⃗=(50),BC⃗=(24),CD⃗=(−50),DA⃗=(−2−4)\vec{AB} = \binom{5}{0}, \quad \vec{BC} = \binom{2}{4}, \quad \vec{CD} = \binom{-5}{0}, \quad \vec{DA} = \binom{-2}{-4}

    Subtract the coordinates at the start of each side from the coordinates at its end.

  2. Compare the vectors of the opposite sides

    AB⃗=−CD⃗,BC⃗=−DA⃗\vec{AB} = -\vec{CD}, \quad \vec{BC} = -\vec{DA}

    AB and CD have the same numbers with opposite signs, so they are parallel and the same length. The same is true of BC and DA.

  3. Deduce that the shape is a parallelogram

    AB∥DC,AD∥BCAB \parallel DC, \quad AD \parallel BC

    Both pairs of opposite sides are parallel, which is exactly what makes a quadrilateral a parallelogram.

  4. Square the length of each side

    AB2=52+02=25,BC2=22+42=20AB^2 = 5^2 + 0^2 = 25, \quad BC^2 = 2^2 + 4^2 = 20

    Pythagoras on the two vectors gives squared lengths of 25 and 20.

  5. Rule out the rhombus

    25≠2025 \neq 20

    The two pairs of sides have different lengths, so the four sides are not all equal and the shape is not a rhombus. It is not a square either, for the same reason.

  6. Test the corner at B for a right angle

    BA⃗⋅BC⃗=(−5)(2)+(0)(4)=−10\vec{BA} \cdot \vec{BC} = (-5)(2) + (0)(4) = -10

    The scalar product of the two sides meeting at B is -10, not 0, so the angle at B is not a right angle.

  7. Rule out the rectangle

    ∠ABC≠90∘\angle ABC \neq 90^\circ

    A rectangle needs all four angles to be right angles, and the angle at B is not one.

  8. Rule out the kite

    AB=DC=5,the equal sides are opposite, not adjacentAB = DC = 5, \quad \text{the equal sides are opposite, not adjacent}

    A kite needs its equal sides to be next to each other. Here the equal sides face each other across the shape, which is the parallelogram pattern, not the kite pattern.

  9. Rule out the trapezium

    2 pairs of parallel sides, not 1\text{2 pairs of parallel sides, not 1}

    A trapezium has exactly one pair of parallel sides. This shape has two, so parallelogram is the more precise name.

  10. Work out the diagonals

    AC⃗=(74),BD⃗=(−34)\vec{AC} = \binom{7}{4}, \quad \vec{BD} = \binom{-3}{4}

    AC runs from (1, 1) to (8, 5) and BD runs from (6, 1) to (3, 5).

  11. Check that the diagonals bisect each other

    mid-point of AC=(4.5,3)=mid-point of BD\text{mid-point of } AC = (4.5, 3) = \text{mid-point of } BD

    Both diagonals have the same mid-point, so they cut each other exactly in half — another property that only the parallelogram family has.

  12. Check the diagonals are not equal

    AC2=72+42=65,BD2=32+42=25AC^2 = 7^2 + 4^2 = 65, \quad BD^2 = 3^2 + 4^2 = 25

    The diagonals have different lengths, which agrees with the shape not being a rectangle.

  13. Check the diagonals are not perpendicular

    7×(−3)+4×4=−21+16=−57 \times (-3) + 4 \times 4 = -21 + 16 = -5

    The scalar product is not zero, so the diagonals do not cross at right angles, which agrees with the shape not being a rhombus.

  14. Collect the evidence

    2 pairs of parallel sides, no right angles, sides not all equal\text{2 pairs of parallel sides, no right angles, sides not all equal}

    The shape is a parallelogram, and none of the extra conditions that would make it a rectangle, a rhombus or a square is satisfied.

  15. State the name of the shape

    Parallelogram\text{Parallelogram}

    Both pairs of opposite sides are parallel and equal, but the sides are not all equal and there is no right angle, so the most precise name is parallelogram.

Answer
Parallelogram\text{Parallelogram}

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