Question

Answer this difficult question of hindustan olympiad 2018

Answer 1

let others angles in all four triangles as w, x, y and z.

refer to attachment ;

=) w + x + y + z = 360°

since w = 180-(a-b), x = 180-(c+d)....similarly for y and z.

=) 180°-(a+b) + 180°-(c+d) + 180°-(e+f) + 180°-(g+h) = 360°

=) 720° - (a+b+c+d+e+f+g+h) = 360°

=) (a+b+c+d+e+f+g+h) = 720-360 = 360°.

Answer 2

Answer: a+b+c+d+e+f+g+h = 360

Step-by-step explanation:

Let w, x, y, z be the Interior angles of the Quadrilateral(IJKL) so formed.

In ΔIMN,

a+b+w = 180 (angle sum property) (w = ∠MIN)

w = 180 - a - b   --------(i)

Similarly,

x = 180 - h - g   -----------(ii)

y = 180 - f - e    -------------(iii)

z = 180 - c - d   -------------(iv)

In Quad IJKL,

w + x + y + z = 360

From (i), (ii), (iii) & (iv)

180 + 180 + 180 + 180 - 360 = a + b + c + d + e + f + g + h

⇒ a + b + c + d + e + f + g + h = 360

Hence, The Sum of the Angles a, b, c, d, e, f, g & h is 360.