Question

prove that the sum of all the angles if a quadrilateral is 360°.

Answer 1

Draw a diagonal inside the quadrilateral such that it divides the quadrilateral into two triangles.(ABC and ADC).Now,According to Angle sum property of triangles,sum of all interior angles in a triangle is 180°.There are two triangles in the quadrilateral so,180+180=360°.
Hence,proved that sum of interior angles in a quadrilateral is 360°.

Answer 2

$\large\green{\boxed{solution : \longrightarrow}}$

Let ABCD be a quadrilateral.

To prove that

∠A +∠B +∠C +∠D = 360°

Const :- Join AC.

Clearly,∠1 + ∠2 = ∠A ...... ( i )

And,∠3 + ∠4 = ∠C ....... ( ii )

Proof :

From Δ ABC , we get

∠2 + ∠4 + ∠B = 180° [By ASP of Δ]...(iii)

From Δ ACD , we get

∠1 + ∠3 + ∠D = 180° [By ASP of Δ]...(iv)

Adding (iii) and (iv) , we get

∠2 + ∠4 + ∠B + ∠1 + ∠3 + ∠D = 360°

=> (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360°

=>∠A + ∠B + ∠C + ∠D = 360° [ using ( i ) and ( ii )

Hence, the sum of all angles of a quadrilateral is 360°.

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \blue{\boxed{ proved}}$

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