If A, B, C and D are four points in a plane such that no three of them are collinear and the line segments AB, BC, CD and DA do not intersect except at their end points, then the closed figure made up of four line segments is called a quadrilateral with vertices A, B, C and D.
Quadrilateral: A plane, closed, geometric figure with four sides.
The word quadrilateral has its origin from the two words "quadric" meaning four and "lateral" meaning sides. Thus, a quadrilateral is that geometrical figure which has four sides, enclosing a part of the plane.
Elements of a Quadrilateral
- Four sides: AB, BC, CD and DA
- Four angles: ∠A, ∠B, ∠C, ∠D
- Two diagonals: AC and BD
- Four vertices: A, B, C and D
Sum of interior angles of a quadrilateral equals 360°.
Sum of exterior angles of a quadrilateral is also 360°.
Example: The three angles of a quadrilateral are 100°, 50° and 70°. Find the measure of the fourth angle.
Types of Quadrilaterals
1. Trapezium: When one pair of opposite sides of quadrilateral is parallel, then it is called a trapezium. If non-parallel sides of a trapezium are equal, then it is called an isosceles trapezium.
2. Kite: When two pairs of adjacent sides of a quadrilateral are equal, then it is called a kite.
3. Parallelogram: When both the pairs of opposite sides of a quadrilateral are parallel, then it is called a parallelogram.
- The opposite sides are equal.
- The opposite angles are equal.
- The diagonals bisect each other.
- Diagonal of a parallelogram divides it into two triangles of equal area.
- Parallelogram on the same base (or equal bases) and between the same parallels are equal in area.
4. Rectangle: It is a special type of parallelogram when one of its angles is right angle.
- Opposite sides are equal.
- Each angle is a right angle.
- Diagonals are equal and bisect each other.
5. Square: When all the four sides of a parallelogram are equal and one of its angles is 90°, then it is called a square.
- All sides are equal.
- Each of the angles measures 90°.
- Diagonals are equal and bisect each other at right angles.
6. Rhombus: When all four sides of a parallelogram are equal, then it is called a rhombus.
- All sides are equal.
- Opposite angles are equal.
- Diagonals of a rhombus are unequal and bisect each other at right angles.
Example: ABCD is a parallelogram. If ∠A = 80°, find the measures of the remaining angles.
Example: Two adjacent angles of a rhombus are in the ratio 4 : 5. Find the measure of all its angles.
Example: One of the diagonals of a rhombus is equal to one of its sides. Find the angles of the rhombus.
Example: The diagonals of a rhombus ABCD intersect at O. If ∠ADC = 120° and OD = 6 cm, find (a) ∠OAD (b) side AB (c) perimeter of the rhombus ABCD.
Mid-Point Theorem
In a triangle the line-segment joining the mid points of any two sides is parallel to the third side and is half of it.
In ∆ABC, if D and E are the mid-points of AB and AC respectively then DE || BC and DE = 1/2 BC.
Other Properties
- The line drawn through the mid point of one side of a triangle parallel to the another side, bisects the third side.
- Triangles on the same base (or equal bases) and between the same parallels are equal in area.
- Triangles on equal bases having equal areas have their corresponding altitudes equal.
Example: D is the mid-point of the side AB of ΔABC and DE || BC. If AC = 8 cm, find AE.
Example: ABCD is a trapezium in which AD and BC are its non-parallel sides and E is the mid-point of AD. EF || AB. Show that F is the mid-point of BC.
Example: ABC is a triangle, in which P, Q and R are mid-points of the sides AB, BC and CA respectively. If AB = 8 cm, BC = 7 cm and CA = 6 cm, find the sides of the triangle PQR.
Equal Intercept Theorem
A line which intersects two or more lines is called a transversal. The line-segment cut off from the transversal by a pair of lines is called an intercept.
If there are three or more parallel lines and the intercepts made by them on a transversal are equal, the corresponding intercepts made on any other transversal are also equal.