So measures80º. Knowingthat angles2 and 3 are also supplementarymeansthat ,
sincethe sum of 100º and 80º is 180º.If angle3 measures100º,then the measureof angle4 mustbe 80º,
since3 and 4 are also supplementary. Noticethat angles1 and 3 are congruent(100º)and 2 and 4 are
congruent(80º).
Theverticalanglestheoremstatesthat if two anglesare verticalanglesthen they are congruent.
We can provethe verticalanglestheoremusinga processjust like the one we usedabove. Therewas
nothingspecialaboutthe givenmeasureof. Hereis proofthat verticalangleswill alwaysbe congruent:
Since and form a linearpair, we knowthat they are supplementary:. For
the samereason, and are supplementary:. Usinga substitution,we can
write. Finally, subtracting on both sidesyields.
Or, by the definitionof congruentangles,.
Use your knowledgeof verticalanglesto solvethe followingproblem.
Example 6
Whatis in the diagrambelow?Usingyour knowledgeof intersectinglines,you can identifythat is verticalto the anglemarked18º. Sinceverticalanglesare congruent,they will havethe samemeasure.So, is also equalto
18º.
LessonSummary
In this lesson,we exploredanglepairs.Specifically, we havelearned:
- How to understandand identifycomplementaryangles.
- How to understandand identifysupplementaryangles.
- How to understandand utilizethe LinearPair Postulate.
- How to understandand identifyverticalangles.