The problem asks us to calculate the value of $x$ given a diagram with several angles around a point on a line. The angles are $x$, $2x$, $19^\circ$, and $172^\circ$. Since the angles form a straight line, their sum must be $180^\circ$.

1. Problem Description

The problem asks us to calculate the value of $x$ given a diagram with several angles around a point on a line. The angles are $x$, $2x$, $19^\circ$, and $172^\circ$. Since the angles form a straight line, their sum must be $180^\circ$.

2. Solution Steps

The angles around a point on a straight line must add up to $180^\circ$. We can set up an equation:

$x + 2x + 19^\circ + 172^\circ = 180^\circ$

Combine like terms:

$3x + 191^\circ = 180^\circ$

Subtract $191^\circ$ from both sides:

$3x = 180^\circ - 191^\circ$

$3x = -11^\circ$

Divide by 3:

$x = \frac{-11^\circ}{3}$

$x \approx -3.67^\circ$

However, looking at the diagram, the angle $x$ should be a positive value. This suggests that the angle $172^\circ$ is not the angle on the straight line.

Let's consider the angles on the other side of the straight line. The angles that make up the full circle are $x$, $2x$, $19^\circ$ and $172^\circ$. Therefore, they should sum up to $360^\circ$. Also, the angle supplementary to the $172^\circ$ is $180^\circ - 172^\circ = 8^\circ$. This $8^\circ$ angle, $x, 2x$ and $19^\circ$ are around the point. Thus, they must add up to $180^\circ$.

So, we have the equation:

$x + 2x + 19 + 8 = 180$

$3x + 27 = 180$

$3x = 180 - 27$

$3x = 153$

$x = 153/3$

$x = 51$

3. Final Answer

$x = 51$