Question

Prove that difference between two sides of a triangle is less

Answer 1

I think the questions should be difference between two side is less than third side I will assume so
How is the difference of any two sides of a triangle less than its third side?



TO PROVE THAT: AC - AB < BC

CONSTRUCTION: From the longer side AC, cut a segment AD = AB

So, now, we need to prove that AC - AD < BC

=>TO PROVE first: that DC < BC

PROOF:< ABD = < ADB = a ( by construction)

< DBC = p, & < BDC = k

Since, k = a + A ( exterior < of a triangle) ……. (1)

p = a - C ( again by exterior < of a triangle) ….(2)

By comparing (1) & (2)

(a + A) > (a - C) , as all these angle variables >0

=> k > p

=> side opposite to k > side opposite to p

=> BC > DC

=> DC < BC

=> AC - AD < BC

=> AC - AB < BC

[ hence proved]