The word quadrilateral is the combination of two Latin words quadri, meaning four, and latus, meaning side. You come across quadrilaterals every day. For example, the page of a book, the top of a pencil box, the top of a dining table and so on, are all quadrilaterals (rectangular shape).
A simple closed figure formed by joining four line segments is called a quadrilateral. It has four sides, four angles, four vertices and two diagonals.
In this chapter, you will learn about basic properties of a quadrilateral.
In a quadrilateral ABCD :
- The four points A, B, C, D are called its vertices.
- The four line segments AB, BC, CD and DA are called its sides.
- ∠DAB, ∠ABC, ∠BCD and ∠CDA are called its angles, to be denoted by ∠A, ∠B, ∠C and ∠D respectively.
- The line segments AC and BD are called its diagonals.
Some important facts about quadrilateral:
- If each angle of a quadrilateral is less than 180°, then it is called convex quadrilateral.
- If each angle of a quadrilateral is greater than 180°, then it is called concave quadrilateral.
Angle Sum Property of quadrilateral:
- Sum of interior angles of quadrilateral is 180°
Let us consider some examples:
Example 1:
The three angles of a quadrilateral are 76°, 54° and 108°. Find the measure fourth angle.
Solution:
We know that sum of the angles of a quadrilateral is 360°.
Let the unknown angle be x
76°+ 54° + 108° + x = 360°.
x = 122°.
Example 2: The angles of a quadrilateral are in the ratio of 3 : 4 : 5 : 6.Find all its angles.
Solution:
Let the angles be 3x°, 4x°, 5x° and 6x°.
3x° + 4x° + 5x° + 6x° = 360°.
18x° = 360°.
x° = 20°.
Hence, the angles are 60°, 80°, 100° and 120°.
Example 3: The angles of a quadrilateral are in the ratio of 4 : 6 : 3. If the fourth angle is 100°, find the other angles of a quadrilateral.
Solution:
Let the angles be 4x°, 6x° and 3x°.
4x° + 6x° + 3x° + 100° = 360°.
13x° = 260°.
x° = 20°.
Hence, the angles are 80°,120° and 60°,
TYPES OF QUADRILATERAL
A closed figure with four sides is a quadrilateral. We come across many different types of quadrilaterals every day. It would be interesting to know the types of quadrilaterals, their shapes and basic properties.
Parallelogram:
A quadrilateral is called a parallelogram, if both pairs of its opposite sides are parallel.
In the figure given below, ABCD is a quadrilateral in which:
AB ∥ DC and AD ∥ BC.
So, ABCD is a parallelogram.
Rhombus:
A parallelogram having all sides equal is called a rhombus.
In the figure given below, ABCD is a rhombus in which:
AB ∥ DC, AD ∥ BC and AB = BC = CD = DA.
Rectangle:
A parallelogram in which each angle is a right angle is called a rectangle.
In the figure given below, ABCD is a quadrilateral in which:
AB ∥ DC, AD ∥ BC and ∠A = ∠B = ∠C = ∠D = 90°.
So, ABCD is a rectangle.
Square:
A parallelogram in which all the sides are equal and each angle measures 90° is called a square.
In the figure given below, ABCD is a quadrilateral in which:
AB ∥ DC, AD ∥ BC, AB = BC = CD = DA.
and ∠A = ∠B = ∠ C = ∠D = 90°.
So, ABCD is a square.
Trapezium:
A quadrilateral having exactly one pair of parallel sides is called a trapezium.
In the figure given below, ABCD is a quadrilateral in which AB ∥ DC. So, ABCD is a trapezium.
If non–parallel sides of a trapezium are equal, it is called an isosceles trapezium.
Kite:
A quadrilateral is called a kite if it has two pairs of equal adjacent sides but unequal opposite sides.
In the figure given below, ABCD is a quadrilateral with AB = AD, BC = DC, AD ≠ BC and AB ≠ DC.
So, ABCD is a kite.
Let us consider an example:
Example: In the square PQRS given in the figure below, PQ = 3x – 7 and QR= x + 3 , find PS.
Solution:
As all sides are equal so, PQ = QR.
3x – 7 = x + 3.
2x = 10.
x = 5.
PQ = 3x – 7 = 8.
QR = x + 3 = 8.
Hence PS = 8.