CK12 - Geometry

If you knowthat the measureof is you can find the measurementof all the otherangles.For

example, and mustbesupplementary(sumto ) becausetogetherthey are a linearpair

(we are usingthe LinearPair Postulatehere).So, to find , subtract from

So,. Knowingthat and are also supplementarymeansthat since

. If , then mustbe because and are also supplemen-

tary. Noticethat (theyboth measure ) and (bothmeasure ). Theseangles
are calledverticalangles. Verticalanglesare on oppositesidesof intersectinglines,and will alwaysbe
congruentby the VerticalAnglesTheorem,whichwe provedin an earlierchapter. Usingthis information,
you can now deducethe relationshipbetweenalternateinteriorangles.

Example 3

Lines and in the diagrambeloware parallel.Whatare the measuresof angles and?

In this problem,you needto find the anglemeasuresof two alternateinterioranglesgivenan exteriorangle.

Use whatyou know. Thereis one anglethat measures Angle correspondsto the angle.So

by the CorrespondingAnglesPostulate,.

Now, because is madeby the sameintersectinglines and is oppositethe angle,thesetwo angles

are verticalangles.Sinceyou alreadylearnedthat verticalanglesare congruent,we conclude

. Finally, compareangles and. Theyboth measure so they are congruent.This will be true any
time two parallellines are cut by a transversal.

We haveshownthat alternateinterioranglesare congruentin this example.Now we needto showthat it
is alwaystrue for any angles.