Essential rules for relationships between sides of a triangle

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

Proven using a geometric argument

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

Relationship between angles and side lengths

The Angle-Side Relationship states that the length of a side is directly related to the magnitude of the angle opposite it

Fundamental for understanding the internal structure of triangles

The Angle-Side Relationship is crucial for comprehending the dimensions and shapes of triangles

Applicable to all triangle types

This theorem is applicable to all types of triangles, including scalene, isosceles, and equilateral

Critical concept in triangle geometry

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

Useful for analyzing relationships between angles and extended sides

This theorem is useful for analyzing the relationships between the angles of a triangle and its extended sides

Determining possible lengths of a third side

The Triangle Inequality Theorem can be used to determine the range of possible lengths for the third side of a triangle

Inferring side lengths from known angles

The Angle-Side Relationship can be used to deduce side lengths when only the angles of a triangle are known