How do we prove the triangle inequality step by step?

Consider the following triangle, ∆ABC: We need to prove that AB + AC > BC. Proof: Extend BA to point D such that AD = AC, and join C to D, as shown below: We note that <ACD = <D, which means that in ∆ BCD, <BCD > <D. Sides opposite larger angles are larger, and thus: BD > BC AB + AD > BC AB + AC > BC (because AD = AC) This completes our proof. We can additionally conclude that in a triangle: Since the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third. The sum of any two sides must be greater than the third side.

Very impressive work. Wish I could do this, too. You know my ultimate goal in terms of math: A. Improve my word problems skills B. Learn Calculus l, ll, and lll at a comfortable level C. Never forget Precalculus. We both know that I am not going to be a teacher. In April I will be 57. I will do my best to keep my math skills sharp. In fact, look for a new PM in about 10 minutes.