Can the following data determine a unique quadrilateral ABCD ?
\[AB = 5{\text{ }}cm,{\text{ }}BC = 6{\text{ }}cm,{\text{ }}CA = 5.2{\text{ }}cm{\text{ }},\]$\angle A = {60^ \circ }and\angle B = {120^ \circ }$ . Justify your answer.

Hint: For solving this particular question we must understand that a quadrilateral has ten parts in all: four sides, four angles and two diagonals. To construct a quadrilateral, we shall need data about five specified parts of it. In this particular question we have five specific parts but not in the required order therefore we cannot determine a unique quadrilateral ABCD with the given data.

Complete step by step solution:
Five measurements can determine a quadrilateral uniquely.
A quadrilateral will be constructed uniquely if the lengths of its four sides and a diagonal are given.
A quadrilateral is often constructed uniquely if the lengths of its three sides and two diagonals are given.
A quadrilateral will be constructed uniquely if its two adjacent sides and three angles are given.
A quadrilateral will be constructed uniquely if its three sides and two included angles are given.
In this question we have three sides that are \[AB = 5{\text{ }}cm,{\text{ }}BC = 6{\text{ }}cm,{\text{ }}CA = 5.2{\text{ }}cm{\text{ }},\] and we have two angles that are $\angle A = {60^ \circ }and\angle B = {120^ \circ }$ , but these are not the included angle, for a unique quadrilateral its three sides and two included angles are required. Therefore, we are not able to construct a unique quadrilateral ABCD.

Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. We divide the desired quadrilateral into two triangles which may be easily constructed. These two triangles together will form a quadrilateral.