Adjacent Angles Investigation: Can Two 90°+ Angles Share a Side?

To solve this problem, let's analyze the properties of the angles involved:

• Definition of obtuse angle: An angle is obtuse if it is greater than $90^\circ$ and less than $180^\circ$.
• Definition of adjacent angles: Adjacent angles share a common side and vertex, and typically form a straight line, summing to $180^\circ$.

Let's consider two adjacent angles, $\angle A$ and $\angle B$, whose sum is $180^\circ$, because they form a straight line.

If $\angle A$ is obtuse, then $\angle A > 90^\circ$.

Similarly, if $\angle B$ is obtuse, then $\angle B > 90^\circ$.

Adding these inequalities, we would have:

$\angle A + \angle B > 90^\circ + 90^\circ = 180^\circ$.

However, since the sum of the angles forming a straight line is exactly $180^\circ$, having both angles greater than $90^\circ$ is impossible as their sum would exceed $180^\circ$. This contradicts the supplementary angle requirement for adjacent angles on a straight line.

Conclusion: Thus, it is not possible to have two adjacent angles that are both obtuse.

Therefore, the answer to the problem is No.