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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 k l , then 2 8 and 3 5 .

Two parallel lines cut by a transversal n, with angles labeled 1 through 8

Proof.

Since k l , by theCorresponding Angles Postulate,

1 5 .

Therefore, by the definition ofcongruent angles,

m 1 = m 5 .

Since 1 and 2 form alinear pair, they aresupplementary, so

m 1 + m 2 = 180 ° .

同时, 5 and 8 are supplementary, so

m 5 + m 8 = 180 ° .

Substituting m 1 for m 5 , we get

m 1 + m 8 = 180 ° .

Subtracting m 1 from both sides, we have

m 8 = 180 ° m 1 = m 2 .

Therefore, 2 8 .

You can prove that 3 5 using the same method.

Theconverseof this theorem is also true; that is, if two lines k and l are cut by a transversal so that the alternate interior angles are congruent, then k l .