Question

Prove the angle sum property of the quadrilateral

Answer 1

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.

Answer 2

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