Before talking about the angle sum property of quadrilaterals, let us recall what a quadrilateral is. A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles. The sum of its interior angles is 360 degrees. 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. In this article, we will take a general quadrilateral and discuss its angle sum property.
Angle Sum Property of a Quadrilateral
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 learnt that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 180°. Let us see how this can be proven.
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.
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