NCERT Solutions for Class 8 Maths Chapter 4 Practical Geometry

There are main 5 types of constructions in Class 8 Maths chapter 4 as per NCERT Book. These are as follows:
When four sides and one diagonal are given.
When two diagonals and three sides are given.
When two adjacent sides and three angles are given.
When three sides and two included angles are given.
When other special properties are known.

No, A quadrilateral can be constructed uniquely if the lengths of its four sides as well as a diagonal is given.

No, a quadrilateral can be constructed uniquely if its three sides as well as two included angles are given.

Yes, it is possible.
A quadrilateral can be constructed uniquely if its two diagonals and three sides are known.

Construct a quadrilateral ABCD in which AB || DC, AB = 5 cm, AD = BC = 6.5 cm and ∠B = 60°.
Steps of construction:
First we draw a rough sketch of quad. ABCD and write down its dimensions as given.
Step 1. Draw AB = 5 cm
Step 2. Make ∠ABX = 60°
(∠C = 1800 – 600 = 1200 because ∠C is supplementary angle of corresponding angle of ∠B)
Step 3. Make ∠BCZ = 120°
Step 4. With C as centre and radius equal to 6.5 cm, draw an arc cutting CZ at point D.
Step 6. Join AD
Then, ABCD is the required quadrilateral,

Draw a rough sketch of the required parallelogram and write down the given dimensions.
Steps of construction:
(i) Draw AB = 6 cm.
(ii) With A as centre and radius 6.8 cm draw an arc.
(iii) With B as centre and radius 4.5 cm draw another arc, cutting the previous arc at C.
(iv) Join BC and AC.
(v) With A as centre and radius 4.5 cm, draw an arc.
(vi) With C as centre and radius 6 cm draw another arc, cutting the previously drawn arc at D.
(vii) Jon DA and DC.
Then, ABCD is the required parallelogram.

We know that the diagonals of a parallelogram bisect each other.
Make a rough sketch of the required parallelogram, as shown.
Steps of construction:
(i) Draw AB = 5.2 cm.
(ii) With A as centre and radius 3.2 cm, draw an arc.
(iii) With B as centre and radius 3 cm draw another arc, cutting the previous arc at O.
(iv) Join OA and OB.
(v) Produce AO to C such that OC = AO and produce BO to D such that OD = OB.
(vi) Join AD, BC and CD.
Then, ABCD is the required parallelogram,

A quadrilateral formed by the points A, B, C and D is complex, if the intersection of AB and CD (if any) lies between the points A and B, and the same applies for BC and DA.

Thus a quadrilateral has 10 elements (4 sides, 4 angles and 2 diagonals) or measurements. To construct a unique quadrilateral, we need to know 5 measurements (elements). Note: To construct a unique quadrilateral simply the knowledge of any five elements is not sufficient.

Since the sum of the exterior angles is always 360 degrees, if you divide 360 degrees by 24 degrees you get 15 which is the number of equal exterior angles and therefore 15 vertices and sides to the polygon.