Quadrilateral: Angle Sum Property

Before talking about the angle sum property of quadrilaterals, let us recall what angles and quadrilateral is. An angle is measured in degrees (°).

A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. In this article, we will take a general quadrilateral and discuss its angle sum property.

Angle Sum Property of a Quadrilateral

Quadrilateral

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

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.

 

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Practise This Question

The regular polygon whose exterior angle is 40 is a: