CAT Alternate Interior Angle Theorem - Practice Questions & MCQ

Theorem 2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Now, using the converse of the corresponding angles axiom, we show the two lines parallel if a pair of alternate interior angles is equal. In the figure, the transversal t intersects lines AB and CD at points E and F respectively such that forming two pairs of alternate interior angles, namely (∠3, ∠5) and (∠4, ∠6).

We will prove that,  ∠3 = ∠5 and ∠4 = ∠6

We have ,

                           ∠3  =  ∠1                 [verticaly opposite angles]

and,                    ∠1  =  ∠5                [Corresponding angles]

Therefore,          ∠3  =  ∠5

Again,                ∠4  =  ∠2                 [verticaly opposite angles]

and,                   ∠4  =  ∠6                [Corresponding angles]

Therefore,         ∠4  =  ∠6

 Hence, ∠3 = ∠5 and ∠4 = ∠6.

Theorem 3: If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

Let's prove this,

We have given, a transversal t cuts two lines AB and CD at E and F respectively such that alternate interior angles is equal, i.e. ∠3 = ∠5.

and we need to prove that the two lines AB and CD are parallel.

We have,          ∠3  =  ∠5            [given]

but,                   ∠3  =  ∠1           [verticaly opposite angle]

therefore,          ∠1  =  ∠5   

But, these are corresponding angles. So, AB is Parallel to CD by correspondig angle axiom.