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Home » Triangles » Triangle Inequalities » The Scalene Inequality Theorem

The Scalene Inequality Theorem

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A scalene triangle is a triangle in which all three sides have different lengths. The scalene inequality theorem states that in such a triangle, the angle facing the larger side has a measure larger than the angle facing the smaller side.

Problem

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

Scalene triangle

Strategy

We have one theorem that tells us when angles in a triangle are equal - the Base Angles Theorem, and one postulate that tells us when one angle is larger than another - the Angle Addition Postulate.

Let's try to combine these two things: to use the Base Angles theorem, we'll construct an isosceles triangle. Since we know AB>AC, we'll find a point D, on side AB, such that AD=AC.

From the Base Angles theorem, we know ∠ADC≅ ∠ACD, and so m∠ADC= m∠ACD. But as D is an interior point to angle ∠ACB, from the angle addition postulate we know that m∠ACB = m∠ACD+m∠DCB, and so m∠ACB > m∠ACD and so m∠ACB > m∠ADC. But ∠ADC is an exterior angle to triangle ΔDBC, and so is equal to the sum of the two remote interior angles.

So if m∠ADC=m∠DBC+m∠DCB, m∠ADC>m∠DBC. Combining this with m∠ACB > m∠ADC we get m∠ACB > m∠ABC

Proof

This is how you prove the Scalene Inequality Theorem:

(1) AB>AC //Given
(2) AD=AC //Construction
(3) ∠ADC≅ ∠ACD //Base Angles theorem
(4) m∠ADC= m∠ACD // Defintion of congruent angles
(5) m∠ACB = m∠ACD+m∠DCB // Angle addition postulate
(6) m∠ACB > m∠ACD //(5), m∠DCB is positive
(7) m∠ADC=m∠DBC+m∠DCB //Exterior angle theorem
(8) m∠ADC>m∠DBC //(7), m∠DCB is positive
(9) m∠ACB > m∠ACD=m∠ADC>m∠DBC //(6), (4), (8)
(10) m∠ACB > >m∠ABC //transitive property of inequality

« Area of a Circle Inscribed in an Equilateral Triangle
Obtuse Triangle - Definition and Properties »

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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