In the image, the problem involves finding the measures of missing angles, and the diagram shows intersecting lines with given angles of 27° and 131°. The task is to find the values for the angles labeled $d$, $e$, and $f$.

Here's how to solve for each:

1. Identifying the relationships between the angles:

   • Vertical angles are equal (angles opposite each other when two lines intersect).
   • Adjacent angles on a straight line are supplementary, meaning they sum to 180°.

2. Step-by-step solution:

   • The angle next to 131° (on the straight line) is supplementary to it. Thus, $f$ can be calculated: $f = 180^\circ - 131^\circ = 49^\circ.$
   • The angle adjacent to $27^\circ$ (on the same straight line) is $d$, and it is supplementary to $27^\circ$: $d = 180^\circ - 27^\circ = 153^\circ.$
   • Since $e$ is vertically opposite to $f$, they are equal: $e = f = 49^\circ.$

So, the missing angles are:

• $d = 153^\circ$,
• $e = 49^\circ$,
• $f = 49^\circ$.

Would you like more details on vertical, adjacent, or supplementary angles, or do you have any further questions?

Here are five related questions:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. Can angles formed by perpendicular lines be supplementary?
3. How do complementary and supplementary angles differ?
4. How can we use the exterior angle theorem in triangle problems?
5. What are co-interior angles and when do they apply?

Tip: Always remember that vertical angles are congruent, and adjacent angles on a straight line sum to 180°.