Construct a quadrilateral ABCD, when \[AB = 4{\rm{cm}}\], \[BC = 5{\rm{cm}}\], \[CD = 6.5{\rm{cm}}\], \[\angle ABC = 105^\circ \] and \[\angle DCB = 80^\circ \]. The length of AD is
A) \[2.9cm\]
B) \[3.5cm\]
C) \[5.5cm\]
D) \[4.5cm\]

Hint:
Here, we will construct a quadrilateral with the given measurements. Then find the length of the line segment AD from the diagram. A quadrilateral is a closed figure with two dimensions, also has four sides and four vertices.

Complete Step by Step Solution:
First We will draw a line segment \[BC = 5cm\].
We will use the following process to construct the quadrilateral:
Now, at B, we will make a line segment BX, which is at an angle of \[105^\circ \] from B using the protractor.
At the line segment BX, we will draw an arc of radius 4 cm on the line segment BX.
We will then mark the point as A.
Now, at C, we will make a line segment CY, which is at an angle of \[80^\circ \] from C using the protractor.
At the line segment CY, draw an arc of radius 6.5 cm on the line segment CY.
We will then mark the point as D.
Now, we will join the points A and D.
Thus, ABCD is a quadrilateral.

Thus by measuring the line segment, we get \[AD = 5.5cm\]
Therefore, the length of AD is \[5.5cm\].

Thus, Option(C) is the correct answer.

Note:
We know that all the geometrical figures are quadrilaterals. Quadrilateral may have all their sides equal, opposite sides equal, parallel sides equal but the angles of the quadrilateral varies from one another except square and rectangle. Thus, we should know that the quadrilateral may be a trapezium, square, rectangle, rhombus, or kite. If we know the three angles of measure of a quadrilateral, we can use the law of sines to find the side of the quadrilateral. But we have only two angles, so this can be performed manually only by constructing the quadrilateral.