, Theseare the givenfacts.
Substitute for usingthe transitive
property.Additionpropertyof equality;add to
both sides.
Arithmetic.Morearithmetic.Substitute for.
Definition.An angleis acuteif and only if its
measureis between and.is an acuteangle.The deductivereasoningschemein example2 is calledaproof. The final statementmustbe true if the
giveninformationis true.
LessonSummary
We built on our previousknowledgeof propertiesof equalityto derivecorrespondingpropertiesof congruence.
This enabledus to test statementsaboutcongruence,and to createnew propertiesand relationshipsabout
congruence.We had our first introduction,in informalterms,to logicalproof.
Pointsto Consider
In the examplesand reviewquestions,termslike given,prove,and reasonwereused.In upcominglessons
we’ll see how to identifythe givenfacts,how to drawa diagramto representa statementthat we needto
prove,and how to organizeproofsmoreformally. As we moveaheadwe’ll provemanyimportantgeometric
relationshipscalledtheorems.We havenow laid the frameworkof logicthat we’ll use repeatedlyin future
work.
LessonExercises
Given: , and are real numbers.
Use the givenpropertyor propertiesof equalityto fill in the blankin eachof the followingquestions.
- Symmetric:If , then ___.
- Reflexive:If , then ___.
- Transitive:If and then __.
- Symmetric:If , then ____.
- Reflexive:If , then _____.
- Substitution:If and , then _____.
- Use the transitivepropertyof equalityto writea convincinglogicalargument(a proof)that the statement
belowis true.
If and and and , then.