CK12 - Geometry

Example 1

If

, then

Proof. Let’s assumetemporarilythat the. Thenwe can reacha contradictionby applying
our standardalgebraicpropertiesof real numbersand equationsas follows:

This last statementcontradictsthe givenstatementthat Hence,our assumptionis incorrectand

we musthave.

We can also employthis kind of reasoningin geometricsituations.Considerthe followingtheoremwhich
we havepreviouslyprovenusingthe CorrespondingAnglesPostulate:

Theorem:If parallellinesare cut by a transversal,thenalternateinterioranglesare

congruent.

Proof. It sufficesto provethe theoremfor one pair of alternateinteriorangles.So consider and.

We needto showthat.

Assumethat we haveparallellines and that. We knowthat lines are parallel,so we have

by postulate that correspondinganglesare congruentand. Sinceverticalanglesare

congruent,we have. So by substitution,we musthave , whichis a contra-
diction.

LessonSummary

In this lessonwe: