Here we will solve the problem on inequalities in triangle.

Let XYZ be a triangle in which XM bisects ∠YXZ. Prove that XY is greater than YM.

Solution:

As XM bisects ∠YXZ, we have ∠YXZ = ∠MXZ ............ (i)

Also, in ∆XMZ, ∠XMY > ∠MXZ, as an exterior angle of a triangle is always greater then either of the interior opposite angles.

Therefore, ∠XMY > ∠YXM, [From (i)].

Therefore, XY > YM, as the greater angle has the greater side opposite to it.

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