Prove that the sum of the angles of the quadrilateral are 360º ?
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A quadrilateral is a polygon which has 4 vertices and 4 sides enclosing 4 angles and the sum of all the angles is 360°. When we draw a draw the diagonals to the quadrilateral, it forms two triangles. Both these triangles have an angle sum of 180°. Therefore, the total angle sum of the quadrilateral is 360°.
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Prove that the sum of all the four angles of a quadrilateral is 360°.
Proof: Let ABCD be a quadrilateral. Join AC.
Clearly, ∠1 + ∠2 = ∠A ...... (i)
And, ∠3 + ∠4 = ∠C ...... (ii)
We know that the sum of the angles of a triangle is 180°.
Therefore, from ∆ABC, we have
∠2 + ∠4 + ∠B = 180° (Angle sum property of triangle)
From ∆ACD, we have
∠1 + ∠3 + ∠D = 180° (Angle sum property of triangle)
Adding the angles on either side, 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 the four angles of a quadrilateral is 360°.
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