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Alternate Interior Angle Theorem
TheAlternate Interior Angles Theoremstates that, when two parallel lines are cut by atransversal, the resultingalternate interior anglesarecongruent.
So, in the figure below, if, thenand.
Proof.
Since, by theCorresponding Angles Postulate,
.
Therefore, by the definition ofcongruent angles,
.
Sinceandform alinear pair, they aresupplementary, so
.
同时,andare supplementary, so
.
Substitutingfor, we get
.
Subtractingfrom both sides, we have
.
Therefore,.
You can prove thatusing the same method.
Theconverseof this theorem is also true; that is, if two linesandare cut by a transversal so that the alternate interior angles are congruent, then.