Question

state and prove alternate interior angle theorem

Answer 1

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The Alternate Interior Angles Theoremstates that, when two parallel lines are cut by a transversal , the resulting alternate interior angles are congruent .

So, in the figure below, if k∥lk∥l , then∠2≅∠8∠2≅∠8 and ∠3≅∠5∠3≅∠5 .



Proof.

Since k∥lk∥l , by the Corresponding Angles Postulate ,

∠1≅∠5∠1≅∠5 .

Therefore, by the definition of congruent angles ,

m∠1=m∠5m∠1=m∠5 .

Since ∠1∠1 and ∠2∠2 form a linear pair , they are supplementary , so

m∠1+m∠2=180°m∠1+m∠2=180° .

Also, ∠5∠5 and ∠8∠8 are supplementary, so

m∠5+m∠8=180°m∠5+m∠8=180° .

Substituting m∠1m∠1 for m∠5m∠5 , we get

m∠1+m∠8=180°m∠1+m∠8=180° .

Subtracting m∠1m∠1 from both sides, we have

m∠8=180°−m∠1           =m∠2m∠8=180°−m∠1           =m∠2 .

Therefore, ∠2≅∠8∠2≅∠8 .

You can prove that ∠3≅∠5∠3≅∠5 using the same method.