1. **State the problem:** We have two intersecting triangles forming an hourglass shape with angles 27°, 35°, 23°, and an unknown angle marked with a question mark opposite the 27° angle. 2. **Identify the key property:** When two lines intersect, opposite (vertical) angles are equal. 3. **Apply the vertical angles theorem:** The angle opposite the 27° angle is also 27° because vertical angles are congruent. 4. **Conclusion:** The unknown angle is $27^\circ$. This is a direct application of the vertical angles theorem, which states that the angles opposite each other when two lines intersect are equal.