In the given image, we have a diagram of intersecting lines with three labeled angles: ∠1, ∠2, and ∠3. The measure of one of the angles is provided as $27^\circ$, which is adjacent to ∠3.

Step-by-Step Explanation:

1. Identify Vertical Angles:

   • ∠1 and ∠3 are vertical angles. Vertical angles are always congruent, meaning they have the same measure.
   • Since ∠3 is adjacent to the given $27^\circ$ angle, we can apply supplementary angle rules to find its measure.

2. Supplementary Angles:

   • Angles on a straight line add up to $180^\circ$. The angle given adjacent to ∠3 is $27^\circ$, so: $\text{∠3} = 180^\circ - 27^\circ = 153^\circ$

3. Vertical Angles:

   • ∠1 and ∠3 are vertical angles, so: $\text{∠1} = \text{∠3} = 153^\circ$

4. Adjacent Angles:

   • ∠2 is adjacent to ∠1, and they are supplementary (forming a straight line). Therefore: $\text{∠2} = 180^\circ - 153^\circ = 27^\circ$

Final Answers:

• ∠1 = 153°
• ∠2 = 27°
• ∠3 = 153°

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How do we prove that vertical angles are always congruent?
2. What is the relationship between adjacent angles and supplementary angles?
3. How would the situation change if the diagram included more intersecting lines?
4. Can two angles be both adjacent and congruent in a similar diagram?
5. How would we find the measures of angles if all we had were algebraic expressions instead of degree values?

Tip: Always remember that angles formed by intersecting lines result in pairs of vertical angles, which are congruent, and adjacent angles that are supplementary.