SPI Mathematics · Free notes

Quadrilaterals: SPI Mathematics

Quadrilaterals has 6 questions in the SPI bank, and one of them is from the 2022 paper. Two are EASY, three MODERATE and one HARD, and each tests a single property: an angle of a parallelogram, the figure made by joining mid-points, or a line cut across a trapezium. The marks go to whoever knows which shape has which property, so this page sets the properties side by side with the diagonals that tell the shapes apart.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

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Subtopic notes

Parallelograms and Other Quadrilaterals
6 questions
The angles of any quadrilateral add to 360 degrees; in a parallelogram opposite sides and angles are equal and neighbouring angles add to 180 degrees.

Formula & revision sheet

3 formulas · 1 reference tables · 8 gotchas across all subtopics — the exam-eve cheat-sheet

Parallelograms and Other Quadrilaterals

Formulas (3)

The angle sum of a quadrilateral · Angles of a quadrilateral $\angle A + \angle B + \angle C + \angle D = 360^\circ \qquad \angle APB = \frac{\angle C + \angle D}{2}$
Sides, angles and height of a parallelogram · Parallelogram $\angle A = \angle C, \ \angle B = \angle D, \ \angle A + \angle B = 180^\circ \qquad h = (\text{slant side}) \times \sin\theta \qquad \text{Area} = \text{base} \times h$
A line parallel to the sides of a trapezium · Line parallel to the bases $\text{length} = a + (b - a)\,t \qquad \text{mid-line} = \frac{a + b}{2}$

Reference tables (1)

Special quadrilaterals and the mid-point figure6 rows

Shape	What makes it	Diagonals	Joining its mid-points gives
Parallelogram	Both pairs of opposite sides parallel	Bisect each other	A parallelogram
Rectangle	A parallelogram with a right angle	Equal, and bisect each other	A rhombus
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles	A rectangle
Square	A rectangle with all sides equal	Equal, and bisect each other at right angles	A square
Kite	Two pairs of equal neighbouring sides	Cross at right angles	A rectangle
Trapezium	Exactly one pair of parallel sides	Cut each other in the ratio of the parallel sides	A parallelogram A trapezium with equal slant sides has equal diagonals, so its mid-point figure is a rhombus.

The mid-point figure depends only on the diagonals: perpendicular gives right angles, equal gives equal sides.

Watch out for (8)

Using 180 for a quadrilateral
→ The angle sum of a quadrilateral
Taking the bisector figure for a rectangle
→ The angle sum of a quadrilateral
Making neighbouring angles equal
→ Sides, angles and height of a parallelogram
Using cos for the height
→ Sides, angles and height of a parallelogram
Thinking the mid-point figure copies the shape
→ Special quadrilaterals and the mid-point figure
Thinking a square is not a rectangle
→ Special quadrilaterals and the mid-point figure
Reading the ratio from the wrong end
→ A line parallel to the sides of a trapezium
Taking the mid-line as half the longer side
→ A line parallel to the sides of a trapezium

Question weightage by concept

4 concepts · 6 questions — where the marks actually sit, so you know what to drill first

Parallelograms and Other Quadrilaterals6 questions · 100%

Concept	Qs	Share
Sides, angles and height of a parallelogram	2	33%
Special quadrilaterals and the mid-point figure	2	33%
The angle sum of a quadrilateral	1	17%
A line parallel to the sides of a trapezium	1	17%

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