To solve for the missing angle $\angle q$ in the triangle, we use the following properties of a triangle:

1. The sum of the angles in any triangle is always $180^\circ$.
2. One angle is given as $45^\circ$, and another angle is a right angle ($90^\circ$).

Steps:

1. Add the known angles: $90^\circ + 45^\circ = 135^\circ$
2. Subtract the sum of the known angles from $180^\circ$ to find $\angle q$: $\angle q = 180^\circ - 135^\circ = 45^\circ$

Final Answer:

The missing angle $\angle q$ is: $m\angle q = 45^\circ$

Let me know if you need further clarification!

Related Questions:

1. How do you classify triangles based on their angles?
2. What is the sum of interior angles for polygons with more than three sides?
3. How do you calculate angles in an isosceles right triangle?
4. How can you identify congruent triangles?
5. What are the properties of a right triangle?

Tip: In a triangle, if two angles are equal (like here), the triangle is also called isosceles!