Triangle inequalities form the basis for understanding the relationships between the sides and angles of triangles. The Triangle Inequality Theorem states that the sum of any two sides of a triangle must exceed the third side's length. This principle, along with the Angle-Side Relationship, which correlates the size of an angle with the length of its opposite side, and the Exterior Angle Inequality Theorem, are essential for geometric reasoning and solving problems when dealing with incomplete data about triangles.
Show More
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must exceed the length of the third side
Construction of arcs with centers at vertices A and B
By constructing arcs with centers at A and B, it can be shown that the direct path along side AB is always shorter than any detour along the arcs
The Angle-Side Relationship states that the length of a side is directly related to the magnitude of the angle opposite it
The Angle-Side Relationship is crucial for comprehending the dimensions and shapes of triangles
This theorem is applicable to all types of triangles, including scalene, isosceles, and equilateral
The Exterior Angle Inequality Theorem states that the measure of an exterior angle is greater than the measure of either of the non-adjacent interior angles
This theorem is useful for analyzing the relationships between the angles of a triangle and its extended sides
The Triangle Inequality Theorem can be used to determine the range of possible lengths for the third side of a triangle
The Angle-Side Relationship can be used to deduce side lengths when only the angles of a triangle are known
00
Triangle Inequality Theorem Expression
|AC| + |BC| > |AB|; |AB| + |BC| > |AC|; |AB| + |AC| > |BC|.
01
Applicability of Triangle Inequalities
Applies to all triangles: scalene, isosceles, equilateral.
02
Purpose of Triangle Inequalities
Determines possible dimensions, ensures triangle formation.
