Properties of special quadrilaterals questions with answers

Questions

1. Question 1Which quadrilateral has these properties?

   Ali describes a quadrilateral. It has every property ticked (✓) in the table, and none of the properties crossed (✗). Write down the special name of the quadrilateral.

   Property	Has it?
   All four angles are right angles.	✗
   All four sides are equal.	✓
   Both pairs of opposite sides are equal.	✓

   Show the worked solution

   1. Step 1: Cross out the shapes where all four angles are right angles✗ Left: rhombus, parallelogram, kite, trapezium, isosceles trapezium.
   2. Step 2: Keep the shapes where all four sides are equal✓ Left: rhombus.
   3. Step 3: Name itIt is a rhombus.
2. Question 2True or false?

   State whether the following statement is true or false. Every rhombus has four right angles.

   Show the worked solution

   1. Step 1: Think of a typical rhombusA rhombus that is not a square does not have four right angles.
   2. Step 2: So the statement isFalse.
3. Question 3Name the shape from its marks

   The marks on the diagram show what is known about quadrilateral $ABCD$. Write down the most special name of $ABCD$.

   Show the worked solution

   1. Step 1: Equal sides (the ticks)Equal sides: two pairs next to each other.
   2. Step 2: Parallel sides (the arrows)Parallel sides: none marked.
   3. Step 3: Right angles (the squares)Right angles: none marked.
   4. Step 4: The most special nameTwo pairs of equal sides next to each other: a kite.
4. Question 4Lengths from the diagonals

   $ABCD$ is a rhombus. Its diagonals meet at $O$. $AC = 12$ cm and $BD = 9$ cm. Find $AB$.

   Show the worked solution

   1. Step 1: The diagonals of a rhombus bisect each other at right angles$OA = 6$ cm, $OB = 4.5$ cm$\angle AOB = 90^\circ$
   2. Step 2: Pythagoras' theorem in triangle AOB$AB^2 = 6^2 + 4.5^2$$AB^2 = 56.25$$AB = \sqrt{56.25}$$AB = 7.5$ cm
5. Question 5Angles from the diagonals

   $ABCD$ is a rhombus. Its diagonals meet at $O$, and $\angle ABD = 47^\circ$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠AOB$\angle AOB = 90^\circ$$\angle ABO = 47^\circ$
   2. Step 2: Angles in triangle AOB$x = 180 - 90 - 47$$x = 43$
6. Question 6Name the quadrilateral from its angles

   The diagram shows a quadrilateral $ABCD$. By finding $x$, name the special quadrilateral $ABCD$.

   Show the worked solution

   1. Step 1: Write an equation in x$(2x + 22)^\circ + (3x + 33)^\circ + (x + 47)^\circ + 108^\circ = 360^\circ$ (∠ sum of quad.)
   2. Step 2: Collect like terms and solve$6x + 210 = 360$$6x = 150$$x = 25$
   3. Step 3: Work out the four angles$\angle A = 72^\circ$, $\angle B = 108^\circ$, $\angle C = 72^\circ$, $\angle D = 108^\circ$
   4. Step 4: What do the angles show?Both pairs of opposite angles are equal (not $90^\circ$): a parallelogram.
   5. Step 5: Name the quadrilateral$ABCD$ is a parallelogram.