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**Hint:**Use the relation that the sum of Reflex angle is $360^\circ $ add all the given values of the angles and substitute all them equals to $360^\circ $ and get the value of angle $QOU$. Use the concepts of lines and angles.

**Complete step by step solution:**

Lines are the ray of infinite length without endpoints. An angle is formed when two lines intersect each other at the same point. The angles are measured in degrees. Angles are of many types. Both the angles and lines concept are used in our daily life.

It is widely used by engineers, architectures to make the designs of roads. An angle is denoted by the symbol $\angle $. An angle has vertex and arms. The common point of an angle is called the vertex of an angle. The two legs forming an angle from a fixed point is called an arm. In the construction of buildings, dams, roads angles are very useful to determine that the design is useful or safe or not.

It is also used in the navy and for pilots. There are many disadvantages of angles such as, bisecting an angle, angles in parallel lines, etc.

Here,

The measure of angle $QOU$is defined as,

\[\angle TOP + \angle POS + \angle SOU + \angle UOQ + \angle QOR + \angle ROT = 360^\circ \]

\[45^\circ + 3x + 55^\circ + 3x + 4x + 3x = 360^\circ \]

\[100 + 13x = 360^\circ \]

\[13x = 360 - 100\]

\[\therefore x = 20^\circ \]

**Hence, the option A is correct.**

**Note:**Always use the formula of sum of angles of a triangle and find the value of an unknown variable. If the sum of two angles is $90^\circ $ then it is called complementary angles and if the sum of two angles is $180^\circ $ then it is called supplementary angles.

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