In the diagram, $\overleftrightarrow{\mathit{O}\mathit{T}}$bisects $\angle SOU$, $m\angle UOV=35$, and $m\angle YOW=120$. Find the measure of each angle.

$m\angle TOU$

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 $\angle A\cong \angle B$.

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, $\angle SOU\cong \angle YOW$.

As, $\angle SOU\cong \angle YOW$, therefore, $\angle SOU=\angle YOW$

Therefore, $\angle SOU=\angle YOW=120°$.

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

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

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

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

$$\begin{gathered} \angle TOU=\frac{1}{2}\times 120° \\ =60° \end{gathered}$$

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