- Definitionof congruentangles
Example 1
Giventhat , whatotheranglesmustbe congruent?Answer: by the RightAngleTheorem,becausethey’reboth right angles.by the Supplementsof the SameAngleTheoremand the LinearPair
Postulate: and are a linearpair, whichmakesthemsupplementary. and
are also a linearpair, whichmakesthemsupplementarytoo. Thenby Supplements
of the SameAngleTheorem, becausethey’resupplementaryto congruent
angles and.Complementsof the SameAngleTheorem
If two anglesare both complementaryto the sameangle(or congruentangles)then the anglesare congruent.
Comments:Onlyone wordis differentin this theoremcomparedto the Supplementsof the SameAngle
Theorem.Herewe haveanglesthat arecomplementary, ratherthansupplementary, to the sameangle.
The proofof the Complementsof the SameAngleTheoremis in the LessonExercises,and it is very similar
to the proofabove.
VerticalAnglesTheorem
VerticalAnglesTheorem:Verticalanglesare congruent.
Verticalanglesare commonin geometryproblems,and in real life whereverlinesintersect:cables,fence
lines,highways,roof beams,etc. A theoremaboutthemwill be useful.The verticalangletheoremis one of
the world’s briefesttheorems.Its proofdrawson the new theoremsjust provedearlierin this section.
Given:Lines and intersect.
Prove: , and