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Home » Triangles » Triangle Inequalities » Converse of the Scalene Triangle Inequality

Converse of the Scalene Triangle Inequality

Last updated: Sep 30, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

In today's lesson, we will prove the converse of the scalene triangle inequality. Using proof by contradiction, we will show that the side facing the larger angle is longer.

Having proven the Scalene Triangle Inequality- that if in a scalene triangle ΔABC, AB>AC then m∠ACB> m∠ABC - proving the converse is very simple.

Problem

In scalene triangle ΔABC, m∠ACB> m∠ABC. Show that AB>AC.

Scalene Triangle

Strategy

When proving converse theorems, it is often useful to use proof by contradiction. What do I mean when I say "proof by contradiction?" I mean we should assume that the converse theorem is NOT true.

Then we will see that in combination with the original theorem (which we know IS true), it leads to a contradiction.

That means the original assumption (that the converse is NOT true) is incorrect. And this is exactly what we will do here.

First assume that AC>AB. By the Scalene Triangle Inequality, we know that if AC>AB, then m∠ABC> m∠ACB. This contradicts what we were given - that m∠ACB> m∠ABC. So AC cannot be larger than AB.

Similarly, if we assume that AC=AB, then by the Base Angle Theorem, m∠ABC= m∠ACB, which contradicts what we were given - m∠ACB> m∠ABC. So AC cannot be equal to AB.

So if AC cannot be larger or equal to AB, it must be smaller. And thus we have proved the converse of the Scalene Triangle Inequality.

« Converse Base Angle Theorem
Hinge Theorem »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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