Any closed polygon with four sides, four angles and four vertices are known as Quadrilateral. It could be a regular or irregular polygon.
If a quadrilateral has one pair of parallel sides then it is a Trapezium.
Here AD∥BC in quadrilateral ABCD, hence it is a trapezium.
Remark: If the non parallel sides of a trapezium are equal then it is called Isosceles Trapezium.
If the two pairs of adjacent sides are equal in a quadrilateral then it is called a Kite.
Here AB = BC and AD = CD
Properties of a kite:
The two diagonals are perpendicular to each other.
One of the diagonal bisects the other one.
∟ A = ∟ C but ∟ B ≠∟ D
If the two pairs of opposite sides are parallel in a quadrilateral then it is called a Parallelogram.
Here, AB ∥ DC and BC ∥ AD, hence ABCD is a parallelogram.
Elements of a Parallelogram:
Some terms related to a parallelogram ABCD
- Opposite Sides: – Pair of opposite sides are
AB and DC, AD and BC
- Opposite Angles: – Pair of opposite angles are
∟ A and ∟ C ∟ B and ∟ D
- Adjacent Sides : – Pair of adjacent sides are
AB and BC BC and DC DC and AD AB and AD
- Adjacent Angles – Pair of adjacent angles are
∠A and ∠B ∠B and ∠C ∠C and ∠D ∠A and ∠D
Properties of a Parallelogram:
- The opposite sides of a parallelogram will always be equal.
Here, AB = DC and AD = BC.
- The opposite angles of a parallelogram will always be of equal measure.
As in the above figure, ∠A = ∠ C and ∠D = ∠B.
- The two diagonals of a parallelogram bisect each other.
Here in ABCD, AC and BD bisect each other at point E.
So that AE = EC and DE = EB.
- The pair of adjacent angles in a parallelogram will always be a supplementary angle.
Example: Three angles of a quadrilateral are in the ratio 3: 4: 5. The difference of the least and the greatest of these angles is 50. Find all the four angles of the quadrilateral.
Solution: The ratio of the three angles of quadrilateral = 3: 4: 5
Let the angles be 3 x, 4 x and 5 x.
The greatest angle among these is 5 x and the least is 3 x.
According to the question, 5 x – 3 x = 50 ⇒ 2 x = 50
⇒ x = 50 ÷ 2 ⇒ x = 25
Hence, the three angles of quadrilateral are 3 × 25 = 75°, 4 × 25° = 100° and 5 × 25° = 125°.
Fourth angle of quadrilateral = 360° – ( 75° + 100° + 125° ) = 360° – 300° = 60°