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Geometry: Theorems

Basic Theorems for Triangles

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Problems

Triangle Inequalities

Triangles are governed by two important inequalities. The first is often referred to as the triangle inequality. It states that the length of a side of a triangle is always less than the sum of the lengths of the other two sides. Can you see why this must be true? Were one side of a triangle longer than the sum of the lengths of the other two, the triangle could not exist. As one side grows, the other two collapse toward that side until the altitude from the vertex opposite the growing side eventually becomes zero. This (an altitude of zero) would happen if the length of the one side was equal to the sum of the lengths of the other two. For this reason, the length of any side must be less than the sum of the lengths of the other sides.

The second inequality involving triangles has to do with opposite angles and sides. It states that when a pair of angles are unequal, the sides opposite them are also unequal. The converse is true also: when a pair of sides are unequal, so are their opposite angles. In essence, this theorem complements the theorem involving isosceles triangles, which stated that when sides or angles were equal, so were the sides or angles opposite them. The theorem about unequal pairs, though, goes a little farther. Given unequal angles, the theorem holds that the longer side of the triangle will stand opposite the larger angle, and that the larger angle will stand opposite the longer side. This inequality is helpful to prove triangles aren't congruent.

Figure %: The larger of two unequal angles is opposite the longer of two unequal sides, and vice versa.
Notice the symbols in the figure above. When angles or sides are equal, the same number of tick marks, or small dashes, can be drawn on them. In a case where sides or angles are unequal, this can be symbolized by different numbers of tick marks on the angles or sides. More tick marks signifies a greater measure.

Exterior Angles of a Triangle

A triangle's exterior angle is just like that of any polygon; it is the angle created when one side of the triangle is extended past a vertex. The exterior angle has two interesting properties that follow from one another. 1) The exterior angle at a given vertex is equal in measure to the sum of the two remote interior angles. These remote interior angles are those at the other two vertices of the triangle. 2) Knowing this, it follows that the measure of any exterior angle is always greater than the measure of either remote interior angle. The first fact (1), the equality, is useful for proving congruence; the second fact (2), the inequality, is useful for disproving congruence.

Figure %: Angle 4 is greater than angle 2 and angle 3; angle 4 = angle 2 + angle 3

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