Quadrilateral properties Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Quadrilateral properties questions. See exactly how to solve problems on angle sum of a quadrilateral, forming an equation, parallelogram, co-interior angles.

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.

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∘360^{\circ}.

  2. Subtract the three known angles

    x=360−285=75x = 360 - 285 = 75

    The three known angles come to 285∘285^{\circ}, and 360360 minus 285285 is 7575.

  3. State the answer

    x=75x = 75

    The fourth angle is 75∘75^{\circ}.

Answer
x=75x = 75
Question 2
2 markseasy
ABCDABCD is a parallelogram. Angle A=74∘A = 74^\circ. Work out the size of angle BB.

Worked solution

  1. Use the angle property of a parallelogram

    ∠A+∠B=180∘\angle A + \angle B = 180^\circ

    ADAD and BCBC are parallel, so angle A and angle BB are co-interior angles between them: they add up to 180∘180^{\circ}.

  2. Subtract from 180180 degrees

    ∠B=180∘−74∘=106∘\angle B = 180^\circ - 74^\circ = 106^\circ

    Taking 74∘74^{\circ} from 180∘180^{\circ} leaves 106∘106^{\circ}.

  3. State the answer

    ∠B=106∘\angle B = 106^\circ

    Angle BB measures 106∘106^{\circ}.

Answer
106∘106^\circ
Question 3
1 markeasy
PQRSPQRS is a parallelogram. Angle Q=65∘Q = 65^\circ. Work out the size of angle SS.

Worked solution

  1. Use the angle property of a parallelogram

    ∠S=∠Q\angle S = \angle Q

    Angle QQ and angle SS are opposite angles of the parallelogram, and opposite angles of a parallelogram are equal.

  2. Write down the size

    ∠S=∠Q=65∘\angle S = \angle Q = 65^\circ

    No calculation is needed: the angle is equal to the one you are given.

  3. State the answer

    ∠S=65∘\angle S = 65^\circ

    Angle SS measures 65∘65^{\circ}.

Answer
65∘65^\circ
Question 4
2 markseasy
ABCDABCD is a rhombus. Angle A=68∘A = 68^\circ. Work out the size of angle BB.

Worked solution

  1. Use the fact that a rhombus is a parallelogram

    ∠A+∠B=180∘\angle A + \angle B = 180^\circ

    A rhombus has four equal sides, which makes its opposite sides parallel — it is a parallelogram. So angles next to each other add up to 180∘180^{\circ}.

  2. Subtract from 180180 degrees

    ∠B=180∘−68∘=112∘\angle B = 180^\circ - 68^\circ = 112^\circ

    Taking 68∘68^{\circ} from 180∘180^{\circ} leaves 112∘112^{\circ}.

  3. State the answer

    ∠B=112∘\angle B = 112^\circ

    Angle BB measures 112∘112^{\circ}.

Answer
112∘112^\circ
Question 5
2 markseasy
ABCDABCD is a kite with AB=ADAB = AD and CB=CDCB = CD. Angle A=108∘A = 108^\circ and angle C=46∘C = 46^\circ. Work out the size of angle BB.

Worked solution

  1. Use the equal angles of a kite

    ∠B=∠D\angle B = \angle D

    In a kite the two angles between the sides of different lengths are equal, so angle BB and angle DD are equal.

  2. Use the angle sum of the quadrilateral

    108+46+2×∠B=360⇒∠B=2062=103108 + 46 + 2 \times \angle B = 360 \Rightarrow \angle B = \frac{206}{2} = 103

    The four angles add up to 360∘360^{\circ}, so the two equal angles share what is left of 360∘360^{\circ} after 108∘108^{\circ} and 46∘46^{\circ} are taken away: 206∘206^{\circ} between them, 103∘103^{\circ} each.

  3. State the answer

    ∠B=103∘\angle B = 103^\circ

    Angle BB measures 103∘103^{\circ}.

Answer
103∘103^\circ

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