Angles in quadrilaterals questions with answers

Questions

1. Question 1Angle sum of a quadrilateral

   The diagram shows a quadrilateral $ABCD$. Find $x$.

   Show the worked solution

   1. Step 1: Add up the four angles of the quadrilateral$96^\circ + 101^\circ + 92^\circ + x^\circ = 360^\circ$
   2. Step 2: Find x$x = 360 - 96 - 101 - 92$$x = 71$ (∠ sum of quad.)
2. Question 2Parallelograms

   $ABCD$ is a parallelogram. Find $x$.

   Show the worked solution

   1. Step 1: Which angle fact gives x?$x^\circ + 54^\circ = 180^\circ$ (int. ∠s, $AB \parallel DC$)
   2. Step 2: Find x$x = 180 - 54$$x = 126$
3. Question 3Rhombuses

   $ABCD$ is a rhombus. Find $x$.

   Show the worked solution

   1. Step 1: Which sides of a rhombus are equal? Use triangle ABC$\angle BCA = 56^\circ$ (base ∠s of isos. △)
   2. Step 2: Add up the angles of triangle ABC$\angle ABC = 180^\circ - 56^\circ - 56^\circ$$\angle ABC = 68^\circ$ (∠ sum of △)
   3. Step 3: A rhombus is a parallelogram: compare ∠ADC with ∠ABC$x = 68$ (opp. ∠s of //gram)
4. Question 4Kites

   $ABCD$ is a kite with $AB = AD$ and $CB = CD$. Find $x$.

   Show the worked solution

   1. Step 1: Which angle of the kite equals x°?$\angle ADC = \angle ABC$$\phantom{\angle ADC} = x^\circ$ ($AC$ is a line of symmetry of the kite)
   2. Step 2: Add up the four angles$64^\circ + x^\circ + 74^\circ + x^\circ = 360^\circ$$2x = 222$$x = 111$ (∠ sum of quad.)
5. Question 5Trapeziums

   $ABCD$ is a trapezium with $AB$ parallel to $DC$. Find $x$.

   Show the worked solution

   1. Step 1: Which angle fact gives x?$x^\circ + 52^\circ = 180^\circ$ (int. ∠s, $AB \parallel DC$)
   2. Step 2: Find x$x = 180 - 52$$x = 128$
6. Question 6Angles as expressions

   The diagram shows a quadrilateral $ABCD$. Find $x$.

   Show the worked solution

   1. Step 1: Write an equation in x$2x^\circ + 70^\circ + (2x + 40)^\circ + (2x - 20)^\circ = 360^\circ$ (∠ sum of quad.)
   2. Step 2: Collect like terms and solve$6x + 90 = 360$$6x = 270$$x = 45$
7. Question 7Angles in a ratio

   In the quadrilateral $ABCD$, $\angle A = 72^\circ$ and $\angle B : \angle C : \angle D = 2 : 3 : 4$. Find $x$.

   Show the worked solution

   1. Step 1: ∠B, ∠C and ∠D together: the angle sum of the quadrilateral less ∠A$\angle B + \angle C + \angle D = 360^\circ - 72^\circ$$\phantom{\angle B + \angle C + \angle D} = 288^\circ$ (∠ sum of quad.)
   2. Step 2: Share it in the ratio$1$ unit $\rightarrow \dfrac{288^\circ}{2 + 3 + 4} = 32^\circ$
   3. Step 3: ∠C is 3 units$x = 3 \times 32$$\phantom{x} = 96$
8. Question 8A square with a diagonal and an isosceles triangle

   $ABCD$ is a square and $AC$ is a diagonal. $E$ is a point inside the square such that $DE = DC$. Find $x$.

   Show the worked solution

   1. Step 1: ∠ADC of the square is 90°: find ∠EDC$\angle EDC = 90^\circ - 44^\circ$$\phantom{\angle EDC} = 46^\circ$
   2. Step 2: DE = DC: triangle DEC is isosceles$\angle DCE = \dfrac{180^\circ - 46^\circ}{2}$$\phantom{\angle DCE} = 67^\circ$ (base ∠s of isos. △, $DE = DC$)
   3. Step 3: ∠BCD of the square is 90°$x = 90 - 67$$x = 23$
9. Question 9A kite with a line parallel to a diagonal

   $ABCD$ is a kite with $AB = AD$ and $CB = CD$. The line $PAQ$ is parallel to the diagonal $BD$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ABD$\angle ABD = \angle PAB$$\phantom{\angle ABD} = 28^\circ$ (alt. ∠s, $PQ \parallel BD$)
   2. Step 2: CB = CD: triangle BCD is isosceles$\angle CBD = \dfrac{180^\circ - 84^\circ}{2}$$\phantom{\angle CBD} = 48^\circ$ (base ∠s of isos. △)
   3. Step 3: Add the two parts$x = 28 + 48$$x = 76$