Quadrilaterals in the Achievement Test

Quadrilaterals in the Achievement Test — Moosa Academy

Achievement test questions on quadrilaterals rarely ask for a proof. They hand you a shape and expect you to recall its properties instantly. This is a revision sheet: six shapes, the properties that separate them, and the traps that cost marks.

Concept The parallelogram — the parent shape

1. Opposite angles are equal

2. Opposite sides are equal and parallel

3. Adjacent angles are supplementary (

180^\circ

)

4. The diagonals bisect each other

5. Each diagonal splits it into two congruent triangles

Learn these five properly. The rectangle, rhombus and square are all parallelograms with extra conditions, so every one of these five carries over to all three.

Concept Rectangle, rhombus, square

Each adds one condition to the parallelogram, and gains one property in return.

A rectangle keeps the sides but squares the corners. A rhombus keeps the angles but equalises the sides. A square does both at once.

Shape	Extra condition	What the diagonals do
Rectangle	All four angles are $90^\circ$	Bisect each other and are equal in length
Rhombus	All four sides are equal	Bisect each other at $90^\circ$ , and bisect the corner angles
Square	Both of the above	Equal, perpendicular, and bisect the corner angles

Concept The isosceles trapezium

1. Exactly one pair of parallel sides (the bases)

2. The two legs are equal in length

3. Each pair of base angles is equal

4. The diagonals are equal in length

5. A leg and a base give supplementary angles

6. The midsegment equals the average of the two bases

Note property 1 says exactly one pair. A trapezium is therefore not a parallelogram, and none of the parallelogram properties may be assumed for it.

Concept The kite

1. Two pairs of equal adjacent sides

2. One diagonal is perpendicular to and bisects the other

3. One pair of opposite angles is equal

4. The main diagonal bisects its two corner angles

5. The diagonals are not equal in length

Property 2 is the one that catches people out. In a kite only one diagonal gets bisected — the main diagonal cuts the other in half but is not itself cut in half.

Note The diagonal test

Most test questions can be settled by asking three things about the diagonals.

Shape	Bisect each other?	Equal?	Perpendicular?
Parallelogram	Yes	No	No
Rectangle	Yes	Yes	No
Rhombus	Yes	No	Yes
Square	Yes	Yes	Yes
Isosceles trapezium	No	Yes	No
Kite	One only	No	Yes

Read down the columns and every shape has a unique pattern. That makes this table the fastest way to identify a shape from a description of its diagonals.

Summary

The parallelogram's five properties are inherited by the rectangle, rhombus and square.
A rectangle adds right angles and gains equal diagonals.
A rhombus adds equal sides and gains perpendicular, angle-bisecting diagonals.
A square has every property of both.
An isosceles trapezium has exactly one pair of parallel sides, equal legs and equal diagonals — but its diagonals do not bisect each other.
A kite has two pairs of equal adjacent sides, and only one of its diagonals is bisected.
When a question describes diagonals, use the comparison table to identify the shape at once.