Whichof the followingis a true biconditionalstatement?A. A polygonis a squareif and only if it has four right angles.B. A polygonis a rhombusif and only if its diagonalsare perpendicularbisectors.C. A polygonis a parallelogramif and only if its diagonalsbisectthe interiorangles.D. A polygonis a rectangleif and only if its diagonalsbisecteachother.Examineeachof the statementsto see if it is true. Beginwith choiceA. It is true that if a polygonis a square,
it has four right angles.However, the conversestatementis not necessarilytrue. A rectanglealso has four
right angles,and a rectangleis not necessarilya square.Providingan examplethat showssomethingis
not true is calledacounterexample.
The secondstatementseemscorrect.It is true that rhombihavediagonalsthat are perpendicularbisectors.
The sameis also true in converse—ifa figurehas perpendicularbisectorsas diagonals,it is a rhombus.
Checkthe otherstatementsto makesure that they are not biconditionallytrue.
The third statementisn’t necessarilytrue. Whilerhombihavediagonalsthat bisectthe interiorangles,it is
not true of all parallelograms.ChoiceC is not biconditionallytrue.
The fourthstatementis also not necessarilytrue. The diagonalsin a rectangledo bisecteachother, but
parallelogramsthat are not rectanglesalso havebisectingdiagonals.ChoiceD is not correct.
So, after analyzingeachstatementcarefully, only B is true. ChoiceB is the correctanswer.LessonSummary
In this lesson,we exploredrhombi,rectangles,and squares.Specifically, we havelearned:
- How to identifyand provethe relationshipbetweenthe diagonalsin a rectangle.
- How to identifyand provethe relationshipbetweendiagonalsin a rhombus.
- How to identifyand provethe relationshipbetweendiagonalsand oppositeanglesin a rhombus.
- How to identifyand explainbiconditionalstatements.
It is helpfulto be able to identifyspecificpropertiesin quadrilaterals.You will be able to use this information
in manydifferentways.
LessonExercises
Use Rectangle
for exercises1-3.