In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

Given: ABCD is a quadrilateral such that DC is the largest side.

To prove: AB + AD > DC - BC

Construction: Join diagonal AC.

Proof: According to the triangle inequality property, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In Δ ABC,

⇒ AB + BC > AC ……………(1)

Similarly, in Δ ADC,

⇒ AD + AC > DC ……………(2)

Adding equation (1) and (2), we get:

⇒ AB + BC + AD + AC > AC + DC

⇒ AB + BC + AD > DC

⇒ AB + AD > DC - BC

Hence, proved AB + AD > DC - BC.