91Ó°ÊÓ

The given diagram is:

The two angles are said to be congruent angles if they have equal measure.

If two angles $A$and $B$are congruent then it can be written as $∠Aâ‰\dots ∠B$.

The theorem 2-3 states that the vertical angles are congruent.

From the diagram it can be noticed that:

The angles $SOU$and $YOW$are vertically opposite angles.

Therefore, from the theorem 2-3, it can be said that the angles $SOU$and $YOW$are congruent angles.

Therefore, $∠SOUâ‰\dots ∠YOW$.

As, $∠SOUâ‰\dots ∠YOW$, therefore, $∠SOU=∠YOW$

Therefore, $∠SOU=∠YOW=120°$.

Therefore, the measure of the angle $SOU$is $120°$.

$\stackrel{↔}{OT}$bisects the angle $SOU$, therefore $∠TOU=\frac{1}{2}∠SOU$.

Therefore, the relation between the angles $SOU$and $TOU$is:$∠TOU=\frac{1}{2}∠SOU$

Find the measure of the angle $TOU$by substituting $120°$for $SOU$in the above relation.

$$\begin{gathered} ∠TOU=\frac{1}{2}×120° \\ =60° \end{gathered}$$

Therefore, the measure of the angle $TOU$is $60°$.

The angles $TOU,UOV$and $ZOT$lies on the straight line $ZV$.

Therefore, the sum of the angles $TOU,UOV$and $ZOT$is $180°$.

Therefore, $∠TOU+∠UOV+∠ZOT=180°$.

As,$∠TOU+∠UOV+∠ZOT=180°$

Therefore,

$$\begin{gathered} ∠TOU+∠UOV+∠ZOT=180° \\ 60°+35°+∠ZOT=180° \\ 95°+∠ZOT=180° \\ ∠ZOT=180°−95° \\ ∠ZOT=85° \end{gathered}$$

Therefore, the measure of the angle $∠ZOT$is $85°$.