## Geometry

# Inequalities In Triangles

## Big Picture

Just as triangles can be congruent or equal they can also be written in inequalities as unequal. Inequalities in triangles are determined by their angle measurements and/or their side lengths.

## Comparing Angles and Sides

*Theorem:* If two sides of a triangle are of unequal length, then the angles opposite these sides are also unequal.

- The largest side is opposite the largest angle, and the shortest side is opposite the shortest side.
- If \(AC > BC > AB\), then \(m \angle B > m \angle A > m \angle C\).

The converse also works: If one angle of a triangle has greater measure than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.

- The largest angle is opposite the longest side, and the smallest angle is opposite the shortest side.
- If \(m \angle B > m \angle A > m \angle C\), then \(AC > BC > AB\).

## Triangle Inequality Theorem

You cannot use any three lengths to make a triangle! The blue arc marks in the figure below show that 4, 5, and 10 cannot make up the lengths of a triangle because the two sides would never meet to form a triangle.

Triangle Inequality Theorem: The sum of the lengths of two sides of a triangle must be greater than the length of the third side.

AB + BC > AC

AC + BC > AB

AB + AC > BC

Tip: Check if the sum of the lengths of the two shorter sides is longer than the third. If it is, the three lengths can make a triangle.

If given two sides of lengths a and b, then the third side has a length c in the range a - b < c < a + b.

## Hinge Theorem

The Hinge Theorem is an extension of the Triangle Inequality Theorem.

*Hinge Theorem (also called Side-Angle-Side Inequality Theorem):* If two sides of two triangles are congruent and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

- \(m \angle B > m \angle E\), so \(AC > DF\)

Converse of the Hinge Theorem: If two sides of two triangles are congruent and the third side of the first triangle is larger than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

- \(AC > DF\), so \(m \angle B > m \angle E\)