CK12 - Geometry

Inequalitiesin Two Triangles

LearningObjectives

• Determinerelationshipsamongthe anglesand sidesof two triangles.


• Applythe SAS and SSS TriangleInequalityTheoremsto solveproblems.

Introduction

In our last lessonwe examinedthe variousrelationshipsamongthe measureof anglesand the lengthsof
the sidesof trianglesand provedthe TriangleInequalityTheoremthat statesthat the sum of the lengthsof
two sidesof a triangleis greaterthan the third side. In this lessonwe will look at relationshipsin two triangles.

SAS InequalityTheorem

Let’s beginour discussionby lookingat the followingcongruenttriangles.

If we thinkof the sidesof the triangleas matchsticksthat are "hinged"at and respectivelythen we

can increasethe measureof the anglesby openingup the sticks.If we openthemso that

, then we see that. Conversely, if you openthemso that , then we see that

. We can provetheoremsthat involvetheserelationships.

SAS InequalityTheorem(The HingeTheorem):If two sidesof a triangleare congruent

to two sidesof anothertriangle,but the includedangleof the first trianglehas greater

measurethan the includedangleof the secondtriangle,then the third side of the first tri-

angleis longerthan the third side of the secondtriangle.

Proof. Consider and with , ,. We

mustshowthat.

Construct so that =. On , take point so that. Either

is on or is not on. In eithercase,we musthave by SAS postulateand