Question

state and prove alternate interior angke theorem

Answer 1

Hllo Frnd____^_^


The Alternate Interior Angles Theorem states 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 .

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

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.

The converse of this theorem is also true; that is, if two lines kk and ll are cut by a transversal so that the alternate interior angles are congruent, then k∥lk∥l .





Hope u understand my Ans.... my friend...___:-)

Answer 2

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