Question

State and prove the angle sum property of a quadrilateral.

Answer 1

The angle sum property :The angle sum property of a quadrilateral states that “the sum of all the angles of a quadrilateral is $360°$.”

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

Proof: In the quadrilateral $ABCD$,

• $\angle ABC,\angle BCD,\angle CDA,$ and $\angle DAB$ 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, $\angle ABC+\angle BCD+\angle CDA+\angle DAB=360°$.

Now consider $∆ADC$,

$\angle D+\angle DAC+\angle DCA=180°$ (Sum of angles in a triangle)

Now consider $∆ABC$,

$\angle B+\angle BAC+\angle BCA=180°$ (Sum of angles in a triangle)

By adding both the equations obtained above we have,

$(\angle D+\angle DAC+\angle DCA)+(\angle B+\angle BAC+\angle BCA)=180°+180°$

$\angle D+(\angle DAC+\angle BAC)+(\angle BCA+\angle DCA)+\angle B=360°$

From figure $\angle DAC+\angle BAC=\angle DAB$

$\angle BCA+\angle DCA=\angle BCD$

$\angle D+\angle DAB+\angle BCD+\angle B=360°$

Which implies,

$\angle D+\angle A+\angle C+\angle B=360°$

Or, the sum of angles of a quadrilateral is $360°$.