CAT Interior Angle Theorem - Practice Questions & MCQ

Theorem 4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

A  transversal t cuts parallel lines AB and CD at point E and F respectively and forming two pairs of consecutive interior angles, namely (∠3, ∠6) and (∠4, ∠5).

We have to prove  ∠3 + ∠6 = 180°  and  ∠4 + ∠5 = 180°

Since ray EF stands on line AB,

we have,          ∠3  + ∠4  = 180°              [linear pair angles]

but,                            ∠4  = ∠6                 [Alternate interior angles]

Therefore,        ∠3  + ∠6  = 180° 

In simmilar way we can show that  ∠4  + ∠5  = 180° 

Theorem 5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

A transversal t, cuts two lines AB and CD at E and F respectively such that ∠4 + ∠5 = 180°.

We need to prove, AB is parallel to CD.

Since ray EB stands on line t,

we have:             ∠1 + ∠4  =  180°                [linear pair angle]

and,                    ∠4 + ∠5  =  180°                [given]

therefore,           ∠1 + ∠4  =   ∠4 + ∠5

this gives,                  ∠1  =  ∠5

But, these are corresponding angles. So AB is parallel to CD.