Question

For a quadrilateral, answer the following question:
(i) How many pairs of adjacent sides does it have?
(ii) How many pairs of opposite sides does it have?
(iii) How many pairs of adjacent angles does it have?
(iv) How many pairs of opposite angles does it have?
(v) How many diagonals does it have?

Answer 1

Hint: In a quadrilateral, two sides are adjacent if they have an angle in common while if two sides don’t have any common angle then they are opposite sides. Similarly two angles are adjacent if they have a side in common while if two angles don’t have any common side then they are opposite angles. Also, a line joining opposite angles is diagonal. Use these results for counting in each case.

Complete step-by-step solution:
According to the question, we have to find the number of adjacent sides, opposite sides, adjacent angles, opposite angles and diagonals in a quadrilateral.

For all the counting, consider the quadrilateral \[ABCD\] shown in the above figure as an example.
(i) We know that in a quadrilateral, two sides are adjacent if they have an angle in common. So if we consider all the four angles in the above figure, we have:
A). For $\angle A$, $DA$ and $AB$ are adjacent sides.
B). For $\angle B$, $AB$ and $BC$ are adjacent sides.
C). For $\angle C$, $BC$ and $CD$ are adjacent sides.
D). And for $\angle D$, $CD$ and $DA$ are adjacent sides.
Thus there are four pairs of adjacent sides.
(ii) We also know that in a quadrilateral if two sides don’t have any common angle then they are opposite sides. Based on this information, we have $AB$ and $CD$ as opposite sides and $BC$ and $DA$ as opposite sides. Thus there are two pairs of opposite sides.
(iii) Further we know that in a quadrilateral two angles are adjacent if they have a side in common. So for the above quadrilateral, we have:
A). For side $AB$, $\angle A$ and $\angle B$ are adjacent angles.
B). For side $BC$, $\angle B$ and $\angle C$ are adjacent angles.
C). For side $CD$, $\angle C$ and $\angle D$ are adjacent angles.
D). And for side $DA$, $\angle D$ and $\angle A$ are adjacent angles.
Thus there are four pairs of adjacent angles.
(iv) Similarly, in a quadrilateral if two angles don’t have any side in common, they are opposite angles. From this we have $\angle A$ and $\angle C$ as opposite angles and $\angle B$ and $\angle D$ as opposite angles. Thus there are two pairs of opposite angles.
(v) And finally, we can see in the figure that opposite angles $\angle A$ and $\angle C$ are joined to form a diagonal $AC$ and $\angle B$ and $\angle D$ are joined to form another diagonal $BD$. Thus a quadrilateral has two diagonals.

Note: The sum of all the angles of a quadrilateral is ${360^ \circ }$. If all the angles measure ${90^ \circ }$ then the quadrilateral is rectangle and if all the sides are equal along with it then the quadrilateral is square. In a rectangle and parallelogram, opposite sides are parallel and equal. And in square and rhombus, opposite sides are parallel and all the sides are equal in length.