Exercise 25 Page 485

According to the Vertical Angle Congruence Theorem, two angles are vertical angles if their sides form two pairs of opposite rays, as shown below.

The doubled marked angles are vertical angles and are, therefore, congruent. Similarly, the single marked angles are vertical angles and are congruent. However, by eyeballing the given diagram, we see that the sides that make up the angles are not opposite rays. Let's exaggerate the picture somewhat to really understand what is happening.

As we can see, $\angle 1$ and $\angle 4$ are certainly not congruent, so our friend is correct.