To find the value of \( x \), we can start by using the property that the sum of the angles in any triangle is \( 180 \) degrees.
For the first triangle, given angles of \( 85 \) degrees and \( 53 \) degrees, we can find the third angle as follows:
\[ \text{Third angle} = 180 - (85 + 53) = 180 - 138 = 42 \text{ degrees}. \]
For the second triangle, the base angles are \( 92 \) degrees and \( x \) degrees. Therefore, we can set up the following equation:
\[ 92 + x + \text{(third angle)} = 180. \]
Let's denote the third angle of the second triangle as \( y \). Thus, we can express it as:
\[ y = 180 - (92 + x) = 180 - 92 - x = 88 - x. \]
Since the two triangles are connected by a common vertex, we can use the information we derived from the first triangle. The third angle of the first triangle (which is \( 42 \) degrees) should equal the third angle of the second triangle:
\[ 88 - x = 42. \]
Now, we can solve for \( x \):
\[ 88 - x = 42 \ -x = 42 - 88 \ -x = -46 \ x = 46. \]
Thus, the value of \( x \) is \( \boxed{46} \).