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°