To find the value of $x$ in the given diagram, let's analyze the triangle and the angles.

Step 1: Analyze the Triangle $ABC$

• The triangle $ABC$ is a right triangle with $\angle B = 45^\circ$ and $\angle C = 50^\circ$.
• The remaining angle, $\angle A$, can be calculated using the sum of angles in a triangle: $\angle A = 180^\circ - \angle B - \angle C = 180^\circ - 45^\circ - 50^\circ = 85^\circ$

Step 2: Analyze Triangle $ADB$

• In the smaller triangle $ADB$, $\angle DAB = 35^\circ$ and $\angle ADB = 45^\circ$.
• We can calculate $\angle ABD$ as: $\angle ABD = 180^\circ - \angle DAB - \angle ADB = 180^\circ - 35^\circ - 45^\circ = 100^\circ$

Step 3: Identify the Angle $x$

• $x$ is the exterior angle to $\triangle ADC$ at point $D$.
• By the exterior angle theorem, $x$ is equal to the sum of the remote interior angles: $x = \angle BDA + \angle ADB = 50^\circ + 35^\circ = 85^\circ$

Thus, $x = 85^\circ$.

Would you like further details, or do you have any questions?

Here are some related questions you might explore:

1. How would you find $x$ if different angles were given?
2. What if the triangle were not a right triangle?
3. How does the exterior angle theorem apply in other situations?
4. What is the significance of the sum of angles in a triangle?
5. How would you verify your solution using trigonometric identities?

Tip: Remember, the sum of the interior angles of any triangle always equals $180^\circ$.