10
Question Will the sum of the angles in a quadrilateral, such as the following one, also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
In the given quadrilateral, join BD. In ∆ABD, we have ∠A + ∠3 + ∠1 = 180° In ∆CBD, we have ∠C + ∠4 + ∠2 = 180° Adding, we get (∠A + ∠3 + ∠1) + (∠C + ∠4 + ∠2) = 180° + 180° ⇒ ∠A + (∠3 + ∠4) + ∠C + (∠1 + ∠2) = 360° ⇒ ∠A + ∠B + ∠C + ∠D = 360° ∴ The sum of the angles of the quadrilateral ABCD is 360°. Also, by using a protractor, we find that the sum of all angles is 360°.