Parallelline Postulate.Hereis a proofof the trianglesum theorem.
- Given: as in the diagrambelow,
Prove: that the measures of the three angles add to or in symbols, that
.
Statement Reason- Given
- Given in the diagram
- Throughpoint , drawthe line parallel2. ParallelPostulate
to. We will call it- AlternateinteriorAnglesTheorem
- AlternateinteriorAnglesTheorem
- AngleAdditionpostulate
- LinearPair Postulate
- Substitution(alsoknownas “transitive
propertyof equality”)
7.
- Substitution(Combiningsteps3, 4, and
7).
8.
And that provesthat the sum of the measuresof the anglesin ANY triangleis
AcuteAnglesin a RightTriangle
Expandingon the trianglesum theorem,you can find morespecificrelationships.Thinkaboutthe implications
of the trianglesum theoremon right triangles.In any right triangle,by definition,one of the anglesis a right
angle—itwill alwaysmeasure This meansthat the sum of the othertwo angleswill alwaysbe
resultingin a total sum of
Thereforethe two acuteanglesin a right trianglewill alwaysbe complementaryand as one of the angles
gets larger, the otherwill get smallerso that their sum is 90°.