GCSE Quadrilateral properties Practice Questions

Use the angle sum of a quadrilateral

The angles of any quadrilateral add up to $360^{\circ}$.

Subtract the three known angles

The three known angles come to $285^{\circ}$, and $360$ minus $285$ is $75$.

State the answer

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

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

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.

Use the clue: Its diagonals cross at right angles

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

State the name of the shape

The quadrilateral described is a kite.

Write down what the question tells you

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

Recall the properties of a rhombus

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 $2$.

Use the diagonal that bisects the angle

The diagonal $AC$ bisects angle $BAD$, so the angle between $AB$ and $AM$ is half of $76^{\circ}$.

Use the perpendicular diagonals

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

Use the angle sum of triangle $ABM$

The three angles of triangle $ABM$ add up to $180^{\circ}$.

State the answer

Angle $ABM$ measures $52^{\circ}$.

Split the parallelogram with its two diagonals

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.

Look at the triangles the diagonals create

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

Work out the angle at a corner

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 $D$.

Deduce that the corner angles are right angles

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 $360 / 4 = 90^\circ$, so the shape is a rectangle.

Rule out the first of the other statements

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

Rule out the second of the other statements

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

Rule out the third of the other statements

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

Rule out the fourth of the other statements

"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 $90^{\circ}$, so it must have right angles.

Check that only one statement survives

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

State the true statement

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

Work out the vector along each side

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

Compare the vectors of the opposite sides

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

Deduce that the shape is a parallelogram

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

Square the length of each side

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

Rule out the rhombus

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.

Test the corner at $B$ for a right angle

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

Rule out the rectangle

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

Rule out the kite

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.

Rule out the trapezium

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

Work out the diagonals

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

Check that the diagonals bisect each other

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

Check the diagonals are not equal

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

Check the diagonals are not perpendicular

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

Collect the evidence

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

State the name of the shape

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.