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}

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}