Free Practical Geometry 02 Practice Test - 8th Grade 

We can't draw a unique quadrilateral with all the measurements which are given. This is because we need the measurement of two adjacent sides and 3 angles to construct the quadrilateral ABCD. But with BC, we will have 6 sets of measurements and two known adjacent sides. Hence, we will be able to construct the quadrilateral ABCD.

If we have the measures of all the sides of a quadrilateral and its one diagonal, we can draw two triangles and hence, form a quadrilateral.

ABCD comprises of two triangle $△$ ADB & $△$ CDB

In Δ ADB, AB = 3cm, BD = 8cm, and AD = 4cm

Since, AB + AD  $<$ BD

$△$ ADB  can't be constructed.

So, quadrilateral ABCD can't be constructed.

Measurement of all the sides or 2 angles and 1 side or 1 angle and 2 sides are required to construct a unique triangle. Using 3 angles, we can construct many triangles, not a unique triangle. For e.g. there are infinite equilateral triangles though they have all the angles equal to ${60}^{\circ }$.

We know that we need minimum 5 measurements from a quadrilateral to draw it.
In one option we have 5 measurements as AB, BC, CD, DA and BD. Here AB, BC, CD, DA are sides and BD is a diagonal.
The construction is possible.

AB, BC, CD, AC and BD. These are 3 sides and 2 diagonals. So, it is possible to construct the quadrilateral.

In other two options only 4 sides and 2 sides are mentioned. So construction is not possible.

Step1: Draw rough diagram of the parallelogram ABCD

Here, $\angle$ DAC = $\angle$  ACB= ${15}^{\circ }$ - (1) (Alternate interior angles)
In $△$ ABC,
$\angle$ A + $\angle$ B + $\angle$ C must be ${180}^{\circ }$
$\angle$ A + $\angle$ B + $\angle$ C = ${25}^{\circ }$ + ${110}^{\circ }$ + ${15}^{\circ }$             (from (1))
$\angle$ A + $\angle$ B + $\angle$ C = ${150}^{\circ }$

The sum of angles in a triangle should always be equal to ${180}^{\circ }$. So, this construction is not possible.

Step 1: Draw AB = 5cm.

Step 2: At B, draw an angle of ${45}^{\circ }$ and mark 6 cm along that angle. Mark the point as C.

Step 3: At C, draw an angle of ${55}^{\circ }$ and mark 7cm along that angle. Mark the point as D.

Step 4: Join point A and D. So, quadrilateral ABCD can be constructed.

Rectangle is a special quadrilateral and minimum number of measures required to construct a rectangle is 2(length of 2 sides).
Since we only know one side of the rectangle. we can draw infinite rectangles. This is because the other side's length is not specified. For e.g.

Let us try to draw the quadrilateral:

Step 1: Draw BC = 6 cm.

Step 2: At B, draw an angle = ${60}^{\circ }$. Along that angle, make an arc of 5 cm and mark it as point A.

Step 3: From A, draw an arc of 6 cm.

Step 4: From C, draw an arc of 7 cm such that it intersects the arc made from A. Mark the point of intersection as D.

Step 5: Join ABCD to complete the quadrilateral.

Is this quadrilateral unique? Yes it is.

If we move it, keeping all the angle length constant $△$ ABC will change. So, there is only one quadrilateral ABCD with AB = 5 cm, BC = 6 cm, CD = 7 cm, BA = 6 cm, $\angle$ ABC= ${60}^{\circ }$.

Since all the sides of a rhombus are equal, it means that if we know one side we know all the sides.

Now,  while constructing the rhombus we need to know at least one angle (as all the angles in a rhombus are not equal). Therefore, in total we need at least 2 measurements to construct a unique rhombus.

Any four measurements are not sufficient to draw a unique quadrilateral.
Let us take AB = 5cm, BC = 4cm, AD = 5cm, CD = 3cm.

We shall start constructing with any one side (say AB). Construct AB with length 5cm. Now, as we have BC and AD. We can draw arcs with BC and AD as radii. But we cannot mark the points C and D based on the data given. So, we need more data to draw a quadrilateral.