Geometry/Chapter 6

The Triangle Inequality

The Triangle Inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side of the triangle.

For instance, if you have a triangle with sides of length A, B and C. Then you know that

• A + B > C
• A + C > B
• C + B > A

In Euclidean Geometry, it is impossible to draw a triangle that violates the Triangle Inequality.

Using the Triangle Inequality

The Triangle Inequality can be used to find the length of a side of a triangle. For instance, say you have a triangle with sides labeled A, B and C. If you know that A = 7, and B = 5 then you can figure out the approximate value of C as follows:

 Statement Reason 7 + 5 > C Triangle Inequality 12 > C Definition of Addition C < 12 Definition of Less Than

So we know that the third side of our triangle is less than 12, but that doesn't help us very much. It turns out that the triangle Inequality can help us again.

 Statement Reason C + 5 > 7 Triangle Inequality C > 7 - 5 Definition of Subtraction C > 2 Definition of Subtraction

In this situation, there is a range of values that C could take. C has to be more than 2, and less than 12. 2 < C < 12.

So it turns out that the measure of the third side of any triangle falls in between the Sum and Differences of the other two sides. Example: If two sides are 9 and 6, then the remaining side's measure would be between 15 (the sum of 9 and 6) and 3 (the difference of 9 and 6). This would result in the following inequality: 3 < c < 15.

There are many triangles that could fit our givens.

Exercises

1) Try to draw a triangle that violates the Triangle Inequality.

2) For each of the following sets of A, B and C, indicate whether the 3 lengths could be the sides of a triangle or not.

• A = 3, B = 4, C = 5
• A = 3, B = 8, C = 5
• A = 14, B = 5, C = 8
• A = 5, B = 5, C = 5

3) Given that you have a triangle with sides of length 12 and 2, write a proof showing what range of values the third side could take. Use a two-column format as shown above.

Triangle Inequality Theorem

In any triangle, the longest side is opposite of the largest angle, the middle sized side is opposite the middle measured angle, and the shortest side is opposite the smallest angle.

Example Problem In triangle ABC, if angle A is 100 degrees and angle B is 35 degrees and angle C is 45 degrees, what are lengths of sides from longest to shortest?

Since side BC is opposite the largest angle, it is the largest side.
Since side AB is opposite the middle sized angle, it is the middle sized side.
Since side AC is opposite the smallest angle, so it is the smallest side.

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Triangle Inequality In Two Triangles: The Hinge Theorem or SAS Inequality Theorem

If two sides of one triangle are congruent to two sides of a second triangle, and the angle between these sides of the first triangle is greater than the angle between these side of the second, then the third side of the first triangle is greater than the third side of the second triangle.

Example: If triangles ABC and DEF have sides AB = DE and
AC = DF and m<A = 80 degrees and m<D = 70 then BC > EF.

Triangle Inequality In Two Triangles: The Converse of Hinge Theorem or SSS Inequality Theorem

If two sides of one triangle are congruent to two sides of a second triangle, and the side between these angles of the first triangle is greater than the side between these angle of the second, then the third angle of the first triangle is greater than the third angle of the second triangle.

Example: If triangles ABC and DEF have the sides
AB = DE BC = EF and AC > DF then, angleB > angleE.