Properties of quadrilaterals and the interior and exterior angles of quadrilaterals

Properties of Quadrilaterals and the Interior and Exterior Angles of Quadrilaterals

9.3

Properties of Quadrilaterals and the Interior and Exterior Angles of Quadrilaterals

The properties of a quadrilateral:

(i) Rectangle

The number of axes of symmetry is \(2\).
The opposite sides are parallel and of equal length.
All of its interior angles are \(90^\circ\).
The diagonals are of equal length and are bisectors of each other.

(ii) Square

The number of axes of symmetry is \(4\).
All the sides are of equal length.
The opposite sides are parallel.
All of its interior angles are \(90^\circ\).
The diagonals are of equal length and are perpendicular bisectors of each other.

(iii) Parallelogram

The number of axes of symmetry is \(0\).
The opposite sides are parallel and of equal length.
The opposite angles are equal.
The diagonals are bisectors of each other.

(iv) Rhombus

The number of axes of symmetry is \(2\).
All the sides are of equal length.
The opposite sides are parallel.
The opposite angles are equal.
The diagonals are perpendicular bisectors of each other.

(v) Trapezium

The number of axes of symmetry is \(0\).
Only one pair of opposite sides is parallel.

(vi) Kite

The number of axes of symmetry is \(1\).
Two pairs of adjacent sides are of equal length.
One pair of opposite angles is equal.
One of the diagonals is the perpendicular bisector of the other.
One of the diagonals is the angle bisector of the angles at the vertices.

Determine the interior angles and the exterior angles of a quadrilateral:

The sum of the interior angles of a quadrilateral is \(360^\circ\).
The sum of an interior angle of a quadrilateral and its adjacent exterior angle is \(180^\circ\).
The opposite angles in a parallelogram (or rhombus) are equal.

Example

The following diagram shows that \(PQRS\) is a parallelogram and \(PST\) is a straight line.

Calculate the value of \(x\) and \(y\).

Noted that the opposite angles in a parallelogram are equal.

So, \(x=42^\circ\).

Also, the sum of an interior angle of a quadrilateral and its adjacent exterior angle is \(180^\circ\).

Thus,

\(\begin{aligned} x+y&=180^\circ \\\\42^\circ+y&=180^\circ \\\\y&=180^\circ-42^\circ \\\\&=138^\circ. \end{aligned}\)