Let's analyze the image.

We are given two intersecting lines, and the measure of one of the angles is $140^\circ$. Angles $\angle 1$ and $\angle 2$ are labeled, and we need to find their measures.

Step-by-step solution:

1. Identify angle pairs:

   • From the diagram, $\angle 1$ and the $140^\circ$ angle are vertical angles, meaning they are equal. Thus, $m\angle 1 = 140^\circ.$

2. Use the Linear Pair Theorem:

   • $\angle 1$ and $\angle 2$ are adjacent and form a straight line, which means they are supplementary. Therefore, their measures add up to $180^\circ$.
   • Using this, we can find $m\angle 2$ as follows: $m\angle 1 + m\angle 2 = 180^\circ,$ $140^\circ + m\angle 2 = 180^\circ,$ $m\angle 2 = 180^\circ - 140^\circ = 40^\circ.$

Final Answer:

• $m\angle 1 = 140^\circ$
• $m\angle 2 = 40^\circ$

Theorem used:

• Vertical Angle Theorem: Vertical angles are congruent (equal).
• Linear Pair Theorem: Adjacent angles that form a straight line are supplementary (add to $180^\circ$).

Let me know if you need further details or clarifications. Here are some follow-up questions you can explore:

1. How do we know that vertical angles are always congruent?
2. Can we use any other theorem to find the measure of $\angle 2$?
3. What is the difference between vertical and supplementary angles?
4. If the given angle were different, how would that affect the measures of $\angle 1$ and $\angle 2$?
5. What if the lines were not straight? Would these theorems still apply?

Tip: Always remember, vertical angles are equal, and adjacent angles on a straight line sum up to $180^\circ$.