Let's break down the given image.

You are asked to find the measures of angles $1$, $2$, and $3$ in the triangle shown. Here's a step-by-step approach:

1. Find Angle 1:

   • The triangle contains a right triangle (since the hint suggests solving angle $1$ first).
   • In the given diagram, the triangle has an angle of $85^\circ$ and $35^\circ$ marked.
   • Since the angles of a triangle sum up to $180^\circ$, the missing angle $1$ is: $\text{Angle 1} = 180^\circ - 85^\circ - 35^\circ = 60^\circ.$

2. Find Angle 2:

   • Using the smaller triangle formed by the division of the original triangle, one of the angles is marked as $35^\circ$. Since it is a right triangle (from the diagram's shape), we can compute the other angle $2$ as: $\text{Angle 2} = 90^\circ - 35^\circ = 55^\circ.$

3. Find Angle 3:

   • From the larger triangle, angle $3$ should complete the right triangle.
   • The angles of a right triangle add up to $90^\circ$. Therefore, angle $3$ is simply: $\text{Angle 3} = 90^\circ.$

Hence, the measures of angles are:

• $\text{Angle 1} = 60^\circ$
• $\text{Angle 2} = 55^\circ$
• $\text{Angle 3} = 90^\circ$

Would you like further details or clarifications?

Here are 5 questions that extend this problem:

1. What is the sum of the interior angles of any triangle?
2. How does the Pythagorean theorem apply to right triangles?
3. Can you use the sine or cosine rule to find angles in this case?
4. What would happen if the triangle was not a right triangle?
5. How do you solve for missing angles in an isosceles triangle?

Tip: Always remember that the sum of the angles in any triangle is $180^\circ$.