Quadrilateral
Quadrilateral means "four sides" (quad means four, lateral means side).So, A closed figure made up of four line segments is called a quadrilateral. It is 2-dimensional (a flat shape) closed (the lines join up) figure made up of 4 straight lines.
The below figure is called a quadrilateral. It can be named in the following ways: ABCD or BCDA or CDAB or DABC
Parts or Elements of a Quadrilateral
1. Four points A, B, C, D are called its vertices.
2. Four line segments AB, BC, CD and DA are called its sides.
3. ∠DAB, ∠ABC, ∠BCD and ∠CDA are called its angles, and can also be written as ∠A, ∠B, ∠C and ∠D respectively. Angles are four in total.
4. The two Line segments AC and BD are called its diagonals. (A line segment joining a pair of opposite vertices is called a diagonal.)
5. The interior angles add up to 360 degrees.
TYPES OF QUADRILATERALS
Parallelogram
A quadrilateral is parallelogram whose opposite sides are parallel and equal in length. Also its opposite angles are equal. Example: rectangle, square and rhombus.
Here, ABCD is a parallelogram and
a. AB ∥ DC and AD ∥ BC.
b. AB = DC; AD = BC
c. ∠A = ∠C; ∠B = ∠D
Rectangle
A rectangle is a parallelogram in which opposite sides are equal in length and each angles measure 90°.
Here, ABCD is a rectangle and
a. AB ∥ DC, AD ∥ BC
b. AD = BC; AB = DC
c. ∠A = ∠B = ∠C = ∠D = 90°
Square
A square is a parallelogram in which all sides are equal in length and each angle measure 90°. It is a rectangle in which all sides are equal.
Here, ABCD is a square and
a. AB ∥ DC, AD ∥ BC
b. AB = BC = CD = DA
c. ∠A = ∠B = ∠ C = ∠D = 90°
Rhombus
A rhombus is a parallelogram whose all sides are equal. A rhombus is sometimes called a diamond.
Here, ABCD is a rhombus and
a. AB ∥ DC, AD ∥ BC
b. AB = BC = CD = DA
c. ∠A = ∠C and ∠ B = ∠D (opposite angles are equal)
Trapezium
A trapezium is a quadrilateral which has a pair of opposite sides parallel.
Here, ABCD is a trapezium and AB ∥ DC
Note: If a trapezium has non parallel sides equal, it is called isosceles trapezium.
Here, ABCD is a isosceles trapezium and
a. AD ∥ BC
b. AB = BC
Sum of Angles of a Quadrilateral
Let us join the opposite vertices of a quadrilateral ABCD.
Now we see two triangles in this figure.
We know that the sum of the angles of a triangle = 180°
As there are two triangles, therefore, the sum of angles of two triangles is 180° + 180° = 360°
So, sum of the angles of a quadrilateral = 360°
Note: No matter what the shape of quadrilateral is, the sum of four angles of a quadrilateral is 360°
Example 1: In a quadrilateral ABCD, ∠A = 80°, ∠B = 105° and ∠C = 115°. Find ∠D.
Solution: We know that the sum of four angles of a quadrilateral ABCD is 360°
Or ∠A + ∠B + ∠C + ∠D = 360°
80° + 105° + 115° + ∠D = 360°
300° + ∠D = 360°
∠D = 360° – 300° = 60°
Example 2: Find the measure of the missing angles in a parallelogram, if ∠A = 60°.
Solution: We know that the opposite angles of a parallelogram are equal, so, ∠C will also measure 60°.
Sum of angles of a quadrilateral = 360°
Or ∠A + ∠B + ∠C + ∠D = 360°
60° + ∠B + 60° + ∠D = 360° (As ∠A = ∠C)
∠B + ∠D + 120° = 360°
∠B + ∠D = 360° – 120° = 240°
But ∠B = ∠D (as opposite angles of a parallelogram are equal)
∠B = ∠D = 240°/2 = 120°