- The proofcan be completedby usingthe AAS postulateto showthat triangles and
are congruent.The conclusionfollowsfrom CPCTC.
Inequalitiesin Triangles
LearningObjectives
- Determinerelationshipsamongthe anglesand sidesof a triangle.
- Applythe TriangleInequalityTheoremto solveproblems.
Introduction
In this lessonwe will examinethe variousrelationshipsamongthe measureof the anglesand the lengths
of the sidesof triangles.We will do so by statingand provinga few key theoremsthat will enableus to de-
terminethe typesof relationshipsthat hold true.
Lookat the followingtwo triangles
We see that the first triangleis isosceleswhilein the the secondtriangle, is longerthan. How
are the measuresof the anglesat and relatedto the lengthsof and It appears(and,in
fact it is the case)that the measureof the angleat vertex is largerthan.
In this sectionwe will formallyprovetheoremsthat revealwhensuchrelationshipshold.We will start with
the followingtheorem.
RelationshipBetweenthe Sidesand Anglesof a Triangle
Theorem:If two sidesof a triangleare of unequallength,then the anglesoppositethese
sidesare also unequal.The largerside will havea largerangleoppositeit.Proof. Consider with. We mustshowthat.- By the RulerPostulate,thereis a point on suchthat. Construct and label
angles , and as follows.