In Fig. 16. 136,

We have,

∠3 = ∠4 (Angle on same segment)

By degree measure theorem,

∠x = 2 ∠3

∠x = ∠3 + ∠3

∠x = ∠3 + ∠4 (i) (Therefore, ∠3 = ∠4)

But,

∠y = ∠3 + ∠1 (By exterior angle property)

∠3 = ∠y - ∠1 (ii)

From (i) and (ii),

∠x = ∠y - ∠1 + ∠4

∠x = ∠y + ∠4 - ∠1

∠x = ∠y + ∠z + ∠1 - ∠1 (By exterior angle property)

∠x = ∠y + ∠z

Hence, proved