Question

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.

$m\angle ZOT$

Short answer

The measure of the angle $\angle ZOT$ is $85°$.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Definition of congruent angles.

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 $\angle A\cong \angle B$.

Step 3. Write the theorem 2-3.

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

Step 4. Write the relations between the angles SOU and YOW.

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

Step 5. Find the measure of the angle SOU.

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 $120°$.

Step 6. Write the relation between the angles SOU and TOU.

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

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

Step 7. Find the measure of the angle TOU.

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 $60°$.

Step 8. Write the relation between the angles TOU, UOV and ZOT.

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, $\angle TOU+\angle UOV+\angle ZOT=180°$.

Step 9. Find the measure of the angle ZOT.

As,$\angle TOU+\angle UOV+\angle ZOT=180°$

Therefore,

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

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