In the figure, lines $l$ and $k$ intersect at point $Q$. We are given the angles:

• $m = 40^\circ$
• $p = 25^\circ$
• $x$ is the unknown angle we need to find.

From the diagram, we observe that $m$, $p$, and $x$ are angles formed at point $Q$, where line $l$ and line $k$ intersect. This creates adjacent angles.

Approach:

• Since $m$ and $x$ are vertical angles, they are equal, meaning $m = x$.
• The angles $p$ and $x$ form a linear pair (on a straight line), so they must sum to $180^\circ$.

Thus, the equation is: $p + x = 180^\circ$ Substitute $p = 25^\circ$ into the equation: $25^\circ + x = 180^\circ$ Solving for $x$: $x = 180^\circ - 25^\circ = 155^\circ$ Since $x$ and $m$ are vertical angles, and we were told that $m = 40^\circ$, then $x = 40^\circ$.

The correct answer is D. 40.

Would you like a detailed breakdown of the solution or have any questions? Here are 5 related questions you can explore:

1. How do you identify vertical angles in any figure?
2. What is the relationship between linear pairs of angles?
3. Can two adjacent angles ever be equal? Under what circumstances?
4. How do you apply angle sum properties in polygons?
5. How do you solve for unknown angles when given multiple intersecting lines?

Tip: When two lines intersect, the opposite (vertical) angles are always congruent!