Quadrilateral: Angle Sum Property

Before talking about the quadrilaterals angle sum property, let us recall what angles and quadrilateral is. The angle is formed when two line segment joins at a single point. An angle is measured in degrees (°).  The quadrilateral is four-sided polygon which can have or not have equal sides. It is a closed figure in two-dimension and has non-curved sides.

A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.  In this article, we will take a general quadrilateral and discuss its angle sum property.

Angle Sum Property of a Quadrilateral

According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees.

Quadrilateral: Angle Sum Property

 

Proof: In the quadrilateral ABCD,

  • ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.
  • AC is a diagonal
  • AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC

We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.

  • We know that the sum of angles in a triangle is 180°.
  • Now consider triangle ADC,

∠D + ∠DAC + ∠DCA = 180°                           (Sum of angles in a triangle)

  • Now consider triangle ABC,

∠B + ∠BAC + ∠BCA = 180°                            (Sum of angles in a triangle)

  • On adding both the equations obtained above we have,

(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°

∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°

  • We see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD.
  • Replacing them we have,

∠D + ∠DAB + ∠BCD + ∠B = 360°

  • That is,

∠D + ∠A + ∠C + ∠B = 360°.

Or, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.

Quadrilateral Angles

A quadrilateral has 4 angles. The sum of its interior angles is 360 degrees. We can find the angles of a quadrilateral if we know 3 angles or 2 angles or 1 angle and 4 lengths of the quadrilateral. In the image given below, a Trapezoid (also a type of Quadrilateral) is shown.

The sum of all the angles ∠A +∠B + ∠C + ∠D = 360°

Quadrilateral Angles

A quadrilateral, in general, has sides of different lengths and angles of different measures. However, squares, rectangles, etc. are special types of quadrilaterals with some of their sides and angles being equal.

Types of Quadrilaterals

There are basically five types of quadrilaterals. They are;

  1. Parallelogram: Which has opposite sides as equal and parallel to each other.
  2. Rectangle: Which has equal opposite sides but all the angles are at 90 degrees.
  3. Square: Which all its four sides as equal and angles at 90 degrees.
  4. Rhombus: Its a parallelogram with all its sides as equal and its diagonals bisects each other at 90 degrees.
  5. Trapezium: Which has only one pair of sides as parallel and the sides may not be equal to each other.

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