Week 13 · Quadrilaterals and polygons: angle sums, special quadrilaterals, interior angles of polygons · 3 lessons
2
Parallelogram, rectangle, rhombus and square: properties and tests
Mathematics Prep 1 Term 1, Unit 4, Lessons 4–5
GoalUse the properties of special quadrilaterals and the tests that show a quadrilateral is a parallelogram.
New words
parallelogram · a quadrilateral whose opposite sides are parallelrectangle · a parallelogram with a right anglerhombus · a parallelogram with four equal sidessquare · a rectangle that is also a rhombus
Explanation
A parallelogram has opposite sides parallel and equal, opposite angles equal, angles next to each other adding up to 180°, and diagonals that bisect each other. A quadrilateral is a parallelogram if any one of these holds: both pairs of opposite sides are parallel; both pairs of opposite sides are equal; one pair of opposite sides is both parallel and equal; the diagonals bisect each other; both pairs of opposite angles are equal. A rectangle is a parallelogram with a right angle and its diagonals are equal. A rhombus is a parallelogram with equal sides; its diagonals meet at right angles and bisect its angles. A square is both: it has every property of a rectangle and a rhombus. Careful: a quadrilateral with equal diagonals is not always a rectangle.
Shape
Diagonals
Angles
Parallelogram
bisect each other
opposite angles equal
Rectangle
equal and bisect each other
all 90°
Rhombus
perpendicular, bisect the angles
opposite angles equal
Square
equal and perpendicular
all 90°
Worked examples
Angles. In parallelogram ABCD angle A = 70°. Opposite angles are equal, so angle C = 70°. Angle B is next to A, so B = 180° − 70° = 110°, and angle D = 110°. Check: 70 + 110 + 70 + 110 = 360.
Diagonals. In parallelogram ABCD the diagonals meet at M. AC = 10 cm and BD = 8 cm. The diagonals bisect each other, so AM = 10 ÷ 2 = 5 cm and BM = 8 ÷ 2 = 4 cm.
A test. A quadrilateral ABCD has AB = CD = 5 cm and BC = DA = 7 cm. Both pairs of opposite sides are equal, so it is a parallelogram. If only AB = BC = 5 cm and CD = DA = 7 cm, the sides next to each other are equal and the test does not apply.
With x. ABCD is a rectangle and its diagonals are AC = (3x + 4) cm and BD = (5x − 6) cm. They are equal: 3x + 4 = 5x − 6, so 2x = 10 and x = 5. Each diagonal is 3 × 5 + 4 = 19 cm.
Class activity
Paper strips. With a grown-up, cross two paper strips of the same width at an angle and glue them: the overlap is a rhombus. Cross two strips of different widths: you get a parallelogram. Measure its sides and diagonals.
Practice
1
In parallelogram ABCD angle D = 65°. Find angles A, B and C.
B = 65° (opposite D). A and C are 180° − 65° = 115° each.
2
The diagonals of a parallelogram are 12 cm and 7 cm. How long are the four parts they are cut into?
The parts of the 12 cm diagonal are 6 cm each; the parts of the 7 cm diagonal are 3.5 cm each.
3
A rhombus has a side of 6 cm and angle A = 64°. Find its perimeter, angle B and the angle that diagonal AC makes with side AB.
Perimeter 4 × 6 = 24 cm. Angle B = 180° − 64° = 116°. The diagonal bisects angle A, so it makes 64° ÷ 2 = 32° with AB.
4
The diagonal AC of a rectangle ABCD is 13 cm. How long is BD? How long is the part of AC from A to the meeting point of the diagonals?
BD = 13 cm because the diagonals of a rectangle are equal. The part is 13 ÷ 2 = 6.5 cm.
5
Sara says: “If the diagonals of a quadrilateral are equal it must be a rectangle.” Give an example that shows she is wrong.
An isosceles trapezium has equal diagonals but it is not a rectangle, because its other pair of sides is not parallel.
