To find the missing angle $a$ in the diagram, let's analyze the situation.

We see that:

• There is a right angle formed by two perpendicular lines.
• One of the angles in this right angle is given as $42^\circ$.
• The missing angle $a$ is adjacent to the $42^\circ$ angle and together with it, they make up the right angle.

Since the sum of angles in a right angle is $90^\circ$, we can calculate $a$ by subtracting $42^\circ$ from $90^\circ$:

$a = 90^\circ - 42^\circ = 48^\circ$

Thus, the missing angle $a$ is $48^\circ$.

Do you need further clarification or details?

Here are some related questions:

1. What is the sum of angles in a triangle, and how does it differ from the sum in a right angle?
2. Can you explain why complementary angles always sum to $90^\circ$?
3. How would you solve for an unknown angle if the lines were not perpendicular?
4. What is the difference between complementary and supplementary angles?
5. How could this problem change if the given angle was an obtuse angle instead of $42^\circ$?

Tip: Always check if angles form a right angle, as it simplifies many angle-finding problems!